IRLF 


571 


•r~\  B  R  A 

u  or  THE 

UNIVERSITY 


PRESENTED     TO    THK    LIBRARY 
OF     THE 

UNIVERSITY 

OF 

CALIFORNIA 


WITH    THE    COMPLIMENTS 
OF    THE 

PUBLISHER 


I  ft 


PLANE  TABLE 


By     JOSEPH      B.      DAVIS 


UNIVERSITY     OF     MICHIGAN 


R 
or  THE 

UNIVERSITY 

^LIFORHit- 


GEORGE    WAH  R 

University     Publisher 
Ann    Arbor,    Mich. 


PLANE  TABLE. 


1.  The  Plane  Table  is  a  drawing  board  in  con- 
nection with  suitable  devices  for  setting,  and  keeping 
it  level.     The  alidade  is  a  straight  edge,  called  the 
rule  and  a  line  of  sight.     The  line  of   sight  is  some- 
times the  edge  of  the  rule,   sometimes  the   line   of 
a    pair   of    compass   sights   set    in  the   line  of  the 
edge   of  the  rule,    or  it  may  be  the  line   of  sight  of 
a  telescope  so  mounted  upon  the  rule  that  this  line  of 
sight  revolves  in  a  plane  at  right  angles  to  the  lower 
face  of  the  rule.     When-a  telescope  is  used,  the  plane 
of  revolution  of  its  line  of   sight  is  set  parallel  to  the 
edge  of  the  rule,  or  to  contain  the  edge  of  the  rule. 

2.  In    the   ordinary  use   of  the   Plane  Table,  a 
piece  of  paper  is  fastened  on  the  board  and  the  lines 
surveyed,  and  points  that  are  determined,  are  platted 
at  once  upon  the  paper.     A  good  quality  of  drawing 
paper  is  commonly  used.     This   paper   may   be   fas- 
tened on  the  board  in  a  variety  of  ways.      If  it   is  to 
remain  there  long,  and  the  board  is   well  shrunk,  it 
may  be  best  to  shrink  it  on  and  stick  the   edges  fast 
with  mucilage.      It   may   also  be  tacked  down  with 
very  small   curtain   tacks.     Thumb  tacks  are  used. 
Clamps  are  used,  and  rollers  beneath  the  board  are 
used  to  some  extent.      If  the  paper  is  larger  than  the 
board   fasten    the   edges  beneath  the  board.      If  the 
paper  is  so  large  as  to  be  shifted  from  time  to  time  to 
receive  the  extended  drawing,  it  may  be  rolled   and 
fastened  beneath  the  board  by  thumb  tacks  or  suit- 


able  clamps.  In  such  a  case  the  part  of  the  paper  on 
the  top  of  the  board  should  be  tacked  fast  to  prevent 
any  shifting.  When  the  paper  is  rolled  under  the 
board,  the  roll,  or  rolls,  and  the  part  of  the  paper 
where  it  passes  over  the  edge  of  the  board  should  be 
protected  by  a  covering  of  stout  paper.  It  is  difficult 
to  keep  the  paper  clean.  When  the  scale  of  the 
map  is  small,  so  that  but  a  small  place  on  the  paper 
needs  to  be  exposed  at  once,  a  newspaper  may  be 
lightly  fastened  over  the  paper  on  the  board  and  torn 
away  in  small  pieces  to  uncover  the  spots  needed 
for  use  at  any  time.  After  a  place  is  finished  and 
no  longer  needed,  it  may  be  covered  again  with 
newspaper,  or  the  newspaper  covering  may  be  changed 
at  times,  and  the  openings  in  it  made  to  fit  the  work. 
In  some  kinds  of  work  an  extra  sheet  of  paper  may 
be  useful  for  laying  down  lines  upon,  or  for  solving 
certain  problems,  connected  with  the  work,  which 
ought  not  to  go  upon  the  map  or  plat. 

3.  Center  of  the  Sheet.  The  plane  table 
should  be  set  up  over  an  instrument  point  in  the 
same  manner  a  transit  is  with  a  plumb  bob  from  its 
center  standing  over  the  mark.  The  observations 
are  then  taken  from  the  center  of  the  sheet.  The 
center  of  the  sheet  is  that  point  on  the  paper  which 
does  not  move  when  the  board  is  turned  about  its 
spindle  axis.  This  point  may  be  found  on  each 
sheet,  or  a  hole  in  the  board  made  to  mark  it.  To 
find  the  center  of  the  sheet  revolve  the  board  on  its 
axis  and  mark  four  fine  marks  on  the  board  near  its 
corners  by  a  pencil  or  needle  held  by  a  fixed  support. 
Find  the  center  of  the  circle  of  which  the  four  marks 
are  a  part  of  the  circumference.  To  do  this  the 


—  5  — 

board  may  be  taken  off  of  its  spindle  and  placed  be- 
side another  board  and  so  longer  ranges  be  obtained 
in  drawing  the  lines  to  the  center.  Draw  all  four 
lines.  They  should  intersect  at  the  same  point.  If 
they  do  not,  try  again.  Having  found  this  center  on 
the  board  itself,  a  piece  of  brass  may  be  set  in  the 
board  with  a  hole  in  it  to  permanently  mark  the  cen- 
ter. When  a  paper  is  fastened  on  the  board,  a  few 
trials  with  a  needle  will  prick  a  hole  in  the  paper  cor- 
responding to  the  hole  in  the  brass  plate.  The 
hole  in  the  brass  plate  may  be  of  the  size  to  receive 
a  common  brass  pin,  which  being  set  firmly  in  the 
hole  may  be  a  very  useful  aid  in  placing  the  alidade, 
when  sighting.  The  center  of  the  sheet  should  be 
found  for  use  on  every  sheet  or  every  time  the  sheet 
is  shifted.  The  hole  in  the  brass  plate  should  be 
tested  for  position  occasionally,  to  see  that  it  stays 
in  its  place.  If,  for  any  reason,  the  connections  of 
the  board  to  the  spindle  are  separated  from  the  board, 
this  test  should  be  made  when  they  are  restored. 

4.  Articles  for  use  with  the  Plane  Table. 

1.  Two  good  triangles,  30°,  60°,  90°,  from  9  to 
12  inches,   according  to  the  size  of  the  board, 
and  the  scale  of  the  map.      These  may  be  of 
metal,  rubber,  celluloid,   or  wood,  but  should 
have  straight  edges.     A  place  to  keep  them  in 
may  be  made  under  the  board. 

2.  Three    legged   dividers,   with    the   third  leg 
running    through    a    tubular    clamp,     so    the 
third  leg  may  be  varied  in  length,  or  removed. 

3.  Lead  pencils,  of  the  grade  of  No.  4  or  5  red 
hexagon,  Faber. 


—  6  — 

4.  Erasers — typewriter,  velvet  rubber,  and  sponge 
rubber. 

5.  Scale  of  equal  parts  for  laying  off  distances. 

6.  Thumb  tacks. 

7.  Curtain  tacks — No.  i. 

8.  A  little  of  Higgins'  Drawing  Board  Mucilage. 

9.  Rubber  cloth.     A  piece   to  cover  the  board 
and  contents  and  fasten  beneath  it  for  protec- 
tion from  rain  and  dirt.     Some  may  prefer  two 
pieces,    or  one   piece   and  a  bag,  the  second 
piece  or  the  bag  being  for  the  alidade. 

10.  A  haversack,  or  better,  a   skeleton   coat  to 
carry  the  various  tools  and  articles  in.      Even 
an   article  on   the   principle  of   a    mechanic's 
apron  with  pockets  on  its  front,  is  good.    The 
skeleton  coat  is  best. 

11.  For   some  work,    a    compass    needle    in   a 
box.     The   box   has   a  square  or  rectangular 
base  with  the  edges  beveled  like  a   rule  and 
parallel  and  perpendicular  to  the  zero  line  of 
the    compass  ring.      The    compass    needle    is 
sometimes  mounted  on  the  rule  of  the  alidade 
but  the  separate  compass  is  better. 

12.  Adjusting  pins,  common  pins,  needle  pricker 
and  reading  glass. 

5.  Setting  up  the  Plane  Table.  Take  the 
table  and  tools  to  an  instrument  point  where  it  is 
desired  that  the  table  shall  be  set  up.  Lay  the 
alidade  and  other  tools  in  a  safe  place.  Tie  on 
the  plumb  bob.  Set  the  table  with  its  tripod,  cen- 
trally over  the  mark  of  the  instrument  point. 
Spread  the  tripod  legs  so  that  when  the  table  is 
finally  set  it  will  come  at  the  elevation  of  the  bend  at 


the  middle  of  the  body  of  the  person  who  is  to  use 
it.  Crowd  the  legs  into  the  eround  till  the  table  will 
stand  firm,  and  the  plumb  bob  center  the  mark. 
The  table  is  usually  set  too  high.  The  arms  are 
rested  on  the  board,  or  the  clothes  rub  against  it, 
thus  stirring  it  from  its  proper  position.  By  setting 
it  down  to  the  bend  in  the  body,  one  can  lean  over 
it  more  conveniently  to  look  through  the  telescope 
or  compass  sights.  Having  attended  to  the  placing 
of  the  table,  stand  the  alidade  centrally  on  the  board. 
Turn  the  board,  or  the  alidade,  till  the  bubble  tubes 
are  parallel  to  the  leveling  screws.  By  operating 
the  leveling  screws  make  the  bubble  tubes  both  read 
level.  Place  the  alidade  centrally  on  the  board  in  a 
new  position.  Level  again,  if  needed.  Proceed  in 
this  way  till  the  table  is  as  perfectly  leveled  as  it 
can  be. 

After  a  little  experience  one  may  leave  the 
board  off  while  setting  up  the  tripod. 

See  that  the  spindle  clamp  and  slow  motion 
screw  are  working  well,  that  the  leveling  screws  are 
all  bearing  somewhat  firmer  than  for  a  transit,  and 
that  the  tripod  head  (in  some  instruments)  is  screwed 
down  tight  to  the  tripod  cap.  Leave  the  spindle  un- 
clamped. 

The  center  of  the  sheet  being  known,  stick  up  a 
common  pin  there.  Everything  being  found,  or 
made,  secure,  and  the  board  standing  steady,  and 
level,  the  plane  table  is  ready  for  work. 

6.  The  line  of  sight  is  set  upon  a  point 
very  much  as  in  the  case  of  a  needle  compass,  or 
transit,  with  the  difference  that  sometimes  the  alidade 
is  moved  on  the  paper  by  the  hands  in  order  to  do  it, 


—  8  — 

and  sometimes  the  whole  instrument  is  moved  about 
its  spindle  axis  and  the  line  of  sight  set  by  the  spindle 
clamp  and  slow  motion  screw,  while  the  alidade 
stands  still  on  the  paper. 

Points  are  located  and  line  given  by  sighting 
with  the  telescope,  the  same  as  with  a  transit,  or 
with  the  compass  sights,  the  same  as  with  a  compass. 

7.  First  work  at  first  station  of  a  survey. 
Set  up  the  table  at  the  station.  Choose  a  point  on 
the  paper  to  represent  the  station  where  the  table 
stands,  so  that  the  rest  of  the  survey  will  come  good 
on  the  paper.  Prick  down  the  point  and  mark  it 
for  that  station.  Choose  the  direction  on  the  paper 
in  which  the  first  line  of  the  survey  is  to  lie,  when 
platted.  Turn  the  board  around  so  the  assumed  di- 
rection of  the  first  line  on  the  paper  is,  by  estima- 
tion, parallel  to  the  line  itself  on  the  ground,  and  the 
same  end  foremost.  Clamp  the  spindle.  Place  the 
working  edge  of  the  rule  of  the  alidade  to  the  pin  at 
the  center  of  the  sheet.  With  the  hands,  keep  the 
rule  to  the  pin,  and  bring  the  line  of  sight  to  look 
to  another  station  of  the  first  line  of  survey.  By 
more  delicate  handling  set  the  line  of  sight  upon  the 
mark  at  the  distant  station.  Place  one  side  of  a 
triangle  against  the  edge  of  the  rule.  Place  the 
other  triangle  to  the  first,  and  slide  the  first  over 
the  paper  till  the  edge  that  was  against  the  rule 
comes  to  the  mark  on  the  paper  for  the  first  station. 
Draw  a  line  through  the  mark  parallel  to  and  in  the 
direction  of  the  line  on  the  ground.  Slide  the  first 
triangle  back  to  the  rule  and  see  if  it  still  fits.  If 
not  review  the  work,  and  correct  the  errors.  Make 
the  line  drawn  of  the  measured,  or  estimated,  length 


—  9— . 

of  the  line  itself,  to  the  scale  of   the  plat,  and  mark, 
at  its  end,  the  second  station  of  the  survey. 

8.  General  directions.     Follow    the  edge  of 
the  rule,  or  triangle,  carefully  with   the  pencil  held  at 
the   same    inclination    to   the    board   for  the   whole 
length  of  the  line.      Keep   the  pencil   sharp.     Draw 
no  more  of   a  line  than  is  needed.      Avoid   erasures, 
as  much  as  possible.      Be  very  particular  when  pro- 
longing a  line  on  the  paper.      Keep  the  edges  of   the 
triangles   straight.     Then  they  may  be  used  to  pro- 
long a  line  by  setting  one  of  them  to  the  line  and  the 
other  to  it  from  the  opposite  side  of   the  line,  when 
they  may  be  slid  past  each  other,  alternately,  in  the 
direction  the  line  is  to  go  to  the  place  the  prolonga- 
tion is  to  reach.     This  operation  may  be    repeated 
and  the  position  of  the    prolongation  tested'.      If  the 
edges  of  the  triangles  are   distrusted  apply  them  to 
the  edge  of  the  metal  rule  of  the  alidade,  and  try  to 
look  through  between  the  parts  in   contact  against  a 
strong  light.      If  the  triangles  are  not  true,  they  may 
be  worked  down  with  emory  or  sand  paper,   or   a 
file,  to  fit  the  edge  of  the   rule.     They  may  also  be 
ground  down  on  a  grindstone. 

9.  Always  place  the   alidade   with    the  rule  on 
the  same  hand   of  the   observer  (either  right  or  left, 
as  is  most  convenient)  on  all  lines  of  the  same  sur- 
vey, when  looking  along    them  in    either   direction. 
Do   not   change  hands  with  the   rule  when  looking 
back  over  a  line.      Keep   the  rule  in  the  same  hand 
and  bring  it  to  the  other  side  of   the  line.      A  right- 
handed  person  will  naturally  take  the  alidade  in  the 
left  hand  and  always  bring  the  rule  to  the  left  side  of 
any  line.     A    left-handed  person  would  reverse  this. 


—  10  — 

This  would  be  the  way  to  place  the  rule  whether 
looking  forward,  or  backward,  along  the  line.  This 
means  that  a  telescope  on  an  alidade  would  not  be 
reversed  on  its  axis  but  remain  the  same  side  up. 

10.  The   two  edges  of  the  rule  should  borh  be 
beveled,    be  straight  and  parallel,  so  that  either  can 
be  used  to  work  from. 

11.  When  carrying  the  table  about,  see  that  the 
board  is  securely  fastened   to  the   tripod  head  so  it 
cannot  fall  off,  and  release  the  spindle  clamp.      The 
plane  table    is    a  clumsy  instrument  to    handle.      It 
must  be  carried  with   the    more    care  and  set   down 
gently.  Otherwise,  the  connections  with  the  spindle  or 
tripod  head  in  many  instruments  can  be  easily  injured 
so  as  to  make    the    table  useless.      The    broad    con- 
nections below  the  board  are  much  the   best.      Carry 
the  alidade,  and  handle  it  by  a  grasp  upon  some  part 
where  a  secure  hold   can    be   obtained,  that  will  give 
good  control  of    it.      Alidades   are    usually  so  heavy 
that  they  should    be    made    with  a  good  hand  hold. 
There  are    so  many  things    to  carry,  some  of  which 
are  awkward  to  handle,    th.it  it  is   best    to    consider 
carefully,  such  features 

12.  Preparations  at  a  second  station  of  a 
survey.     These  are  to  place   the   board    in    position 
for  making  observations' at  a  station  by  means    of    a 
line  platted  on  the  paper  which  runs  to   that  station. 
It  is  orienting  the  board  by  means  of  a    platted  line. 

Set  up  the  table  at  the  station.  Take  out  the 
pin  at  the  center  of  the  sheet.  Place  the  edge  of  a 
triangle  to  the  platted  line.  Slide  the  triangle  by 
means  of  the  other  as  if  to  draw  a  line  at  the  cen- 
ter of  the  sheet,  nearly  covering  the  center  of  the 


—  11- 
sheet.  Hold  the  first  triangle  down  steady  with  the 
fingers  near  the  edge  on  the  center  of  the  sheet. 
With  the  other  hand,  bring  the  edge  of  the  rule  of 
the  alidade  up  to  the  edge  of  the  triangle  at  the  cen- 
ter of  the  sheet.  With  a  sliding  movement  of  the 
alidade  fit  the  edge  of  the  rule  to  the  triangle.  Be 
careful  not  to  shift  the  triangle,  as  it  is  easily  moved 
by  the  heavier  alidade.  After  setting  the  rule  see  if 
it  stands  parallel  to  the  line  it  was  set  by. 

If  the  spindle  happens  to  be  clamped,  release  it. 
Turn  the  board  to  direct  the  line  of  sight  to  the  dis- 
tant station  on  the  line  used  in  placing  the  alidade 
at  the  center  of  the  sheet.  Clamp  the  spindle.  By 
the  spindle  slow  motion  screw  set  the  line  of  sight  on 
the  mark  at  the  distant  station. 

In  placing  the  triangle  to  the  line  in  the  first 
place,  and  the  alidade  at  the  center  of  the  sheet,  have 
regard  to  the  relative  position  of  things  on  the  table 
and  on  the  ground,  so  that  when  the  line  of  sight  is 
finally  set  on  the  mark  at  the  distant  station,  the  line 
joining  the  plats  of  the  stations  on  the  board  shall  be 
parallel  to  the  corresponding  line  on  the  ground  and 
lie  the  same  end  foremost. 

The  board,  or  table,  now  stands  parallel  to  its 
former  position.  It  is  oriented,  and  ready  for  work. 

13.  Measuring  a  vertical  angle.  Set  up  the 
table.  Place  the  alidade  central  on  the  table,  with 
the  edge  of  the  rule  at  the  center  of  the  sheet  and  the 
line  of  sight  on  a  mark  at  the  station  on  the  line  from 
which  the  vertical  angle  is  to  be  measured.  Set  the 
horizontal  cross  wire  on  the  mark  that  shows  one 
side  of  the  angle  to  be  read,  by  means  of  the  tele- 
scope axis  clamp  and  slow  motion  screw.  Read  the 


—  12  — 

vertical  circle.  Record  the  reading.  Release  the 
telescope  axis.  By  the  means  used  before  set  the  hori- 
zontal cross  wire  on  the  mark  on  the  other  side  of  the 
angle.  Read  the  vertical  circle.  Record  the  read- 
ing. Release  the  telescope  axis.  By  the  same 
means  set  the  horizontal  cross  wire  again  upon  the 
first  mark.  Read  the  vertical  circle.  Record  the 
reading.  See  if  the  first  and  last  readings  agree  suffi- 
ciently well.  If  not,  review  the  work  and  correct 
the  errors.  The  difference  of  the  readings  will  show 
the  angle. 

14.  If  the  vertical  angle  is  an  angle  of  elevation 
or  depression,  set  the  horizontal  cross  wire  on  the 
mark  as  first  above  directed.  Read  the  vertical  cir- 
cle. Record  the  reading.  Release  the  telescope 
axis.  By  its  clamp  and  slow  motion  screw  set  the 
telescope  so  its  level  will  read  level.  Read  the  verti- 
cal circle.  Record  the  reading.  Reverse  the  alidade 
on  the  board  and  bring  it  up  to  the  line  with  the  rule 
on  the  opposite  side  of  the  center  of  the  sheet.  Set 
the  telescope  level  to  read  level,  if  it  needs  it,  by 
means  of  its  slow  motion  screw.  Read  the  vertical 
circle.  Record  the  reading.  Release  the  telescope 
axis.  Set  the  horizontal  cross  wire  on  the  mark 
again  by  means  of  the  clamp  and  slow  motion  screw 
to  the  telescope  axis.  Read  the  vertical  circle. 
Record  the  reading.  See  if  the  two  middle  readings 
(taken  when  the  telescope  was  level)  agree  well.  If 
they  are  very  different,  the  board  is  not  flat,  or  it  is 
unsteady.  It  may  be  set  firmer,  but  if  it  is  uneven, 
the  angle  value  will  be  poor.  Having  found  the 
readings  to  be  as  good  as  practicable,  find  the  angle  by 
the  first  pair  of  readings,  and  also  by  the  second  pair. 


—  13  — 

Take  the  half  sum  of  the  two  values  for  the  correct 
one. 

15.  To  measure  the  angle  of  elevation  or 
depression  of  a  line,  measure  the  height  of  the 
telescope  axis  from  the  ground  and  sight  to  a  mark 
at  the  same  height  from  the  ground  at  the  other  end 
of  the  line.  If  the  line  is  not  on  the  ground,  meas- 
ure the  distance  of  the  telescope  axis  from  it  in  a 
similar  manner,  and  sight  to  a  mark  at  a  correspond- 
ing distance  from  the  line  at  another  place  on  it. 
Measure  the  angle  of  elevation  or  depression  of  the 
line  from  the  transit  axis  of  the  telescope  to  the  cor- 
responding mark  at  the  station  by  the  method  de- 
scribed. 

16-  Rem.    The  work  that  can  be  done  with  the 
plane  table  depends  directly  upon  the  methods  of  as- 
certaining the   position  of   a  point   with   respect   to 
other  known  points.     These  methods  come  first. 

The  expression  "  set  the  alidade"  upon  a  line  or 
mark  means  that  the  edge  of  the  rule  is  to  be  placed 
at  the  center  of  the  sheet  and  the  line  of  sight  set 
upon  that  line,  or  mark. 

Capital  letters  denote  points  on  the  ground,  and 
the  small  letters  the  plats  of  the  same  points  on  the 
sheet. 

The  work  of  the  plane  table  can  be  done  with  a 
transit  and  drawing  board,  by  platting  the  work  as  it 
goes  on. 

17-  Determining  a  point  by  direct  meas- 
urement.    Orient  the  table  at  a  known  station.    Set 
the  alidade  upon  the  line  to  the  required  point.     Plat 
a  line  from  a  plat  of  the  instrument  station,  parallel 
to  the  edge  of  the  rule  and  running  towards  the  re- 


—  14  — 

quired  point.  Measure  the  distance  from  the  instru- 
ment station  to  the  required  point,  by  chain,  steel 
tapes,  or  stadia.  If  the  stadia  is  used,  the  telescope 
axis  should  stand  at  the  center  of  the  sheet.  Lay 
off  this  distance,  to  the  proper  scale,  from  the  station 
on  the  plat  along  the  line  drawn.  Prick  down  the 
point  thus  reached,  and  mark  it  for  the  required 
point. 

18.  Determining  a  point  by  intersections. 
Fig.  1.     Let  A  and  B  be  determined  and  platted  at 
Set  up  the  table  at  A  and  orient  it  by  a  b. 
Set   the    alidade   on  a    line   to 
C.      Draw  from  a,  towards  C,  a 
line  parallel  to   the   edge  of  the 
rule,  and  continue  it  to  the  esti- 
mated position  of  c  on  the  sheet. 
Set  the  alidade  again  upon  A  B. 
See   if   the   edge  of   the  rule  is 

f  A     / 

o  parallel  to  a  b.     If  not,  review 

the  work  and  correct  the  errors.  Take  up  the 
plane  table.  Set  it  up  at  B.  Orient  it  by  b  a. 
Set  the  alidade  on  B  C,  a  line  to  C.  Draw  from  b, 
towards  C,  a  line  parallel  to  the  edge  of  the  rule  till 
it  intersects  the  line  from  a,  parallel  to  A  C.  Set 
the  alidade  again  upon  A  B.  See  if  the  edge  of  the 
rule  is  parallel  to  b  a.  If  not,  review  the  work  and 
correct  the  errors.  Prick  down  the  intersection  of 
the  two  lines  drawn  from  a  and  b  and  mark  this 
point  c. 

But  a  short  piece  of  a  c,  or  b  c,  or  of  both,  may 
be  needed,  near  c.  Any  number  of  points  may  be 
determined  from  a  base  A  B.  a  c  and  b  c  should  in- 
tersect as  nearly  as  may  be  at  right  angles.  If  a  c 


—  15  — 

and  b  c  intersect  at  a  small  angle,  the  result  cannot  be 
trusted.  Other  lines  to  c  from  d,  c,  or/,  other  known 
points,  will  help  to  fix  its  position.  A  number  of 
lines  to  c  \vi\\  not  all  intersect  at  the  same  point  on 
the  paper.  The  proper  location  of  the  point  on  pa- 
per is  theoretically  a  question  of  mathematical  ad- 
justment, but  there  are  usually  some  circumstances 
connected  with  the  work  which  will  enable  the  selec- 
tion to  be  made  properly. 

19.  Determining  points  by  resection.  This 
is  in  reality  a  method  by  intersections.  The  differ- 
ence between  this  and  the  last  case  is,  that  one  of  the 
two  lines  necessary  to  determine  the  point  is  drawn 
on  the  paper,  while  the  table  is  set  up  at  the  point 
whose  position  is  sought.  In  the  previous  case  ob- 
jects may  be  located,  or  determined,  without  going 
to  them.  In  the  case  of  resection  the  table  is  set  up 
at  the  station,  or  object,  whose  position  £is  sought. 
This  is  an  advantage  in  carrying  out  work,  as  the  ta- 
ble may  be  advanced  to  a  station  before  it  is  on  the 
map,  if  there  is  a  line  on  the  paper  that  runs  to  it 
from  some  known  station  that  is  in  sight  from  it.  By 
referring  to  Fig.  i  the  principle  of  this  method  may 
be  stated.  Let  A  and  B  be  known  and  platted,  at  a 
and  b.  Set  up  the  table  at  A  and  orient  it  by  a  b. 
Set  the  alidade  on  a  line  to  C.  Draw  from  a,  to- 
wards C,  a  line  parallel  to  the  edge  of  the  rule,  and 
continue  it  to  where  c  is  thought  to  be  on  the  paper, 
or  beyond.  Set  the  alidade  again  upon  A  B.  See  if 
the  edge  of  the  rule  is  parallel  to  a  b.  If  not,  review 
the  work,  and  correct  the  errors.  Take  up  the  plane 
table.  Set  it  up  at  C.  Orient  it  by  the  indefinite  line 
from  a,  towards  c.  Set  the  alidade  on  C  B.  Draw 


—  16  — 

from  b  (towards  C  were  the  table  at  B),  a  line  inter- 
secting the  indefinite  line  from  «,  and  parallel  to  the 
edge  of  the  'rule.  Set  the  alidade  again  upon  C  A. 
See  if  the  edge  of  the  rule  is  parallel  to  the  indefinite 
line  from  a  towards  c.  If  not,  review  the  work,  and 
correct  the  errors.  In  a  similar  manner  lines  may  be 
drawn  on  the  map,  or  plat,  from  any  number  of  sta- 
tions in  sight  -from  C,  whose  positions  have  been 
platted.  Several  lines  thus  drawn,  running  to  c,  will 
help  fix  its  position  with  much  greater  precision. 

20.  Hem.  A  single  intersection  is  not  sufficient 
to  fix  a  station  on  the  plat.  A  direction  line  and  a 
measured  distance  are  not  alone  sufficient.  When  a 
point  is  to  be  determined  by  intersection,  there  should 
be  a  number  of  lines  observed  that  run  to  it.  When 
a  point  is  to  be  determined  by  measurements,  there 
should  be  two  or  more.  A  combination  of  intersect- 
ing lines  and  measurements  may  be  used.  The  po- 
sition of  any  station  on  the  plat  should  not  be  used  to 
determine  the  position  of  any  other  station,  or  object, 
unless  its  own  position  has  been  well  checked. 
Every  available  check  upon  the  position  of  a  station 
should  be  used.  Neglect  no  chance  to  tie  up  a 
station.  A  check  which  shows  an  error  in  the  position 
of  the  station  tested,  may  really  reveal  an  error  in  the 
position  of  the  station  from  which  the  test  comes.  A 
check  works  at  both  ends,  and  tests  the  position  of 
both  stations  concerned.  What  constitutes  a  good 
determination  of  a  point?  This  can  be  ascertained 
only  by  experience  upon  any  particular  work,  and 
with  the  instruments  in  use.  The  stadia  may  be  used 
with  the  best  plane  table  to  get  distances  and  heights 
no  much  work.  For  general  topographical  surveys 


—  17  — 

the  stadia  will  probably  give  good  results.  When 
filling  in  final  details  to  a  large  scale,  it  may  be  found 
unsuitable,  especially  as  to  elevations.  Any  general 
statement  relating  to  such  matters  must  be  materially 
modified  by  the  person  doing  the  work  and  the  cir- 
cumstances of  the  case.  These  facts  render  such 
statements  of  little  value. 

21.  In  orienting   the  table  use  more  than  a  sin- 
gle line,    where  practicable,  no  matter  how  well  that 
line  may  have  been  determined.     That  is,   orient  by 
a  line,  and  test  the   orientation  by   sights   to  other 
stations,  if  possible.     This  serves  to  further  check  the 
plat. 

22.  The  advantage  of  resection  is,  that  after' one 
or  more  lines  to  the  new  station  have  been  platted,  it 
is  possible  to  set  up  the  table  at  that  station,  and  by 
observing  to  other  stations  get  a  number  of  lines  run- 
ning to  the  new  station  with  a  single  setting  of  the  ta- 
ble,  when  by  intersections,  each   of  the  stations  ob- 
served would  have  to  be  occupied.     This   is   often  a 
very  great  advantage.     With  some  patience,  it  is  con- 
ceivable that  the  table  might  be   oriented  at   a   new 
station  by  persistent  trial  in  taking  pointings  on  other 
stations,  at   the   same   time  the  new  station  was  de- 
termined.    There  are  better  ways  of  doing  this,  given 
below. 

23.  Determining  a  point  by  the   "Three 
Point  Problem"  and  orienting  the  table.    Fig. 

2.  Let  the  three  points,  A,  B,  and  C,  be  well  de- 
termined upon  the  plat.  Let  the  position  of  D,  from 
which  A,  B,  and  C,  can  be  seen,  be  required,  without 
going  to  them. 


—  18  — 

F*asten  an  extra  piece  of  paper  over  the  middle  of 
the  board.  Find  the  center  of  the  sheet  on  this  pa- 
per. Call  the  center  -of  the  sheet  on  this  paper  m. 

Set  up  the  table  at  D. 
Clamp  the  table  so  the 
extra  piece  of  paper  will 
receive  the  lines  D  A,  D 
B,  and  D  C,  when  platted 

from    the    center     of    the 
* 
sheet.      Set  the  alidade  on 

DA  and  draw  a  line,  by 

fig  2  t^ie  ectee  °f  tne  rule>  from 

m    towards    A.      Set    the 

alidade  on  D  B  and  D  C,  in  succession,  and  draw 
lines  from  m  towards  B  and  C,  respectively.  Notice 
the  position  of  D  with  respect  to  A  B  C,  that  is,  see 
if  D  lies  on  the  convex  or  concave  side  of  the  broken 
line  ABC.  Fit  the  three  legged  dividers  to  the 
platted  positions,  on  the  plane  table  sheet,  of  the 
points  A,  B,  C.  Make  the  fit  carefully.  Apply 
the  points  of  the  dividers  to  the  lines  from  m 
towards  A,  B,  and  C,  each  point  of  the  dividers 
upon  the  line  running  towards  the  point  it  was 
fitted  to  the  plat  of.  By  careful  adjustment,  make 
the  points  of  the  dividers  fit  the  three  lines. 
Prick  down  the  points  on  the  lines  where  the 
dividers  fit.  The  center  of  the  sheet,  m,  on  the 
extra  piece  of  paper,  is  in  the  same  relative  posi- 
tion with  respect  to  the  three  points,  on  the  three 
lines,  that  were  pricked  down  with  the  points  of  the 
three  legged  dividers,  that  D  is  with  respect  to  A,  B, 
and  C,  on  the  ground,  and  may  be  transferred  to  the 
plane  table  sheet  with  the  three  legged  dividers, 


-19  — 
thus   obtaining   the   correct  plat   of   all    four  points. 

Orient  the  table  by  means  of  a  line  from  the 
plat  of  D,  on  the  plane  table  sheet,  to  the  plat  of 
either  A,  B,  or  C,  and  test  its  position  by  lines  to  the 
other  points.  Use  the  longest  line  to  orient  by. 

When  the  plats  of  A,  B,  and  C  cannot  be  em- 
braced by  the  three  legged  dividers,  use  the  same 
equal  parts  of  the  lines  from  the  plat  of  B  to  the  plats 
of  A  and  C,  and  take  up  the  plat  of  B,  and  the  two 
auxiliary  points  with  the  dividers.  Fit  the  dividers 
to  the  three  lines  from  m  as  before.  The  point  m 
will  form  the  fourth  angle  of  a  quadrilateral  whose 
sides  bear  the  same  ratio  to  the  sides  of  the  one  re- 
quired on  the  plat,  that  the  parts  first  used  to  do  the 
whole  lines  used  on  the  plat.  Fix  a  third  auxiliary 
point  on  the  plat  with  respect  to  the  plat  of  B  and 
the  two  first  auxiliary  points  with  the  dividers.  Draw 
auxiliary  lines  from  this  third  auxiliary  point  to  the 
others  and  to  the  plat  of  B.  Extend  the  line  to  the 
plat  of  B  beyond  the  third  auxiliary  point.  Draw 
lines  from  the  plats  of  A  and  C,  respectively  parallel 
to  the  near  sides  of  the  quadrilateral  of  auxiliary  lines 
and  produce  them  to  intersect.  They  should  inter- 
sect on  the  line  from  the  plat  of  B.  If  they  do  not, 
review  the  work,  and  correct  the  errors.  This  in- 
tersection is  the  plat  of  D. 

24.  Determining  a  point  by  the  "Two 
Point  Problem,"  and  orienting  the  table.  Fig. 
3.  Let  two  points,  A  and  B,  be  well  determined 
on  -the  plat.  Let  the  position  of  C,  from  which  A 
and  B  can  be  seen,  be  required,  without  going  to 
them.  This  is  possible  when  a  fourth  point  D  can 
be  occupied  which  is  in  sight  from  C,  and  from 


—  20  — 

which  A,  B,  and  C,  can  be  seen.  D  may  be 
made  use  of  in  continuing  the  survey,  or  not.  As- 
sume a  station,  D. 

Fasten  an  extra  piece  of  paper  over  the    center 

of  the  board.     Find  the  center  of  the   sheet   on   this 

paper,  and  call  it  m.      Set  up 

f- -7?^  the   table  at    D.     Clamp    the 

•  \,  s''    i       table  so  the  extra  paper  will 

I          'xvX          ;        receivea  plat   of   the   quadri- 
>     I         /  I        lateral    A    B    C    D.    Set    the 

;  /''  X ;         alidade  on  D  C,  and  draw  a 

jj  C    line,  by  the  edge  of  the  rule, 

from  m  towards  C.     Set  the 
F'$  3  alidade  on  D  A  and  D  B,  in 

succession,  and  draw  lines  from  m  towards  A  and  B 
respectively.  Remove  the  table  to  C,  and  set  it  up 
there.  Orient  the  table  by  the  line  from  m  towards 
C.  Assume  any  convenient  length  from  m  on  this 
line  to  represent  the  length  of  D  C,  and  prick 
down,  on  the  extra  paper,  a  temporary  plat  of 
C.  Set  the  alidade  on  C  A  and  C  B,  in  succes- 
sion, and  complete,  upon  the  assumed  length  for 
D  C,  a  temporary  plat  of  A  B  C  D.  By  this 
means  obtain  a  temporary  plat  of  A  B  upon  the 
extra  piece  of  paper.  The  angle  between  the  plat  of 
A  B  on  the  plane  table  sheet  and  the  temporary  plat 
of  A  B  on  the  extra  sheet  is  the  amount  the  board  is 
to  be  turned  to  correctly  orient  it  with  respect  to  A 
B  on  the  ground  and  the  plat  of  A  B  on  the  plane 
table  sheet.  Set  the  alidade  at  m  parallel  to  the 
temporary  plat  of  A  B  upon  the  extra  sheet. 
Find  or  fix  a  mark  in  range  of  the  line  of  sight. 
Set  the  alidade  at  m  parallel  to  the  correct  plat 


—  21  — 

of  A  B  on  the  plane  table  sheet.  Release  the 
spindle  clamp.  Set  the  line  of  sight  on  the  mark  by 
means  of  the  spindle  clamp  and  slow  motion  screw. 
The  table  is  oriented  correctly  with  respect  to  A  B. 
Resect  upon  A  and  B  to  determine  C. 

After  orienting  the  table  its  position  may  be 
tested.  Set  the  alidade  on  m  and  parallel  to  the  tem- 
porary plat  of  either  C  D  or  C  B  on  the  extra  sheet. 
Release  the  spindle  clamp.  Set  the  line  of  sight  on 
D,  or  B,  by  means  of  the  spindle  clamp  and  slow 
motion  screw.  Set  the  alidade  on  m  and  parallel  to 
the  temporary  plat  of  A  B  on  the  extra  sheet.  The 
line  of  sight  should  strike  the  mark  formerly  used. 

Particular  notice  must  be  taken  to  observe  which 
side  of  A  B,  the  station  C  is  on  the  ground  and  to 
conform  the  temporary  plat  to  the  position  of  the 
stations  on  the  ground  so  as  to  prevent  reversing  the 
table  when  finally  orienting  it,  or  of  using  the  supple- 
ment of  the  proper  angle  when  making  that  orien- 
tation. 

If  D  is  to  be  used  in  continuing  the  survey  ;  after 
resecting  upon  A  and  B  to  get  C,  set  the  alidade  on 
C  D  and  draw  a  line  from  the  plat  of  C  to  get  the 
plat  of  D.  Move  the  table.  Set  it  up  at  D.  Orient 
it  by  the  line  from  the  plat  of  C  towards  D.  Resect 
on  A  and  B  to  get  D.  It  should  fall  on  the  line  from 
the  plat  of  C. 

The  quadrilateral  on  the  plane  table  sheet,  or 
the  triangle  A  B  C,  in  case  D  is  not  platted  on  the 
plane  table  sheet,  may  be  compared  with  the  tempo- 
rary plat  upon  the  extra  sheet.  The  angles  should 
be  equal,  and  the  sides  proportional. 


—  22  — 

C  D  may  lie  in  any  relation  to  A  B — cross   it   if 
necessary. 

25.  When  it  is  possible  to  set  the  table  in   line 
with  A  B,  by  trial,  as  at  D,  Fig.  4,    orient  it    by   a  b 
A       at  once.      Set   the   alidade  on  D  C 
and  draw  a  line  towards  C,  anywhere 
on  the  sheet.      Move  the  table  to  C. 
Set    it    up.       Orient   it    by    the    last 
named  line.      Resect  on  A  and  B  to 
get    C.      Set    the  alidade   on   C    D. 
jj  Draw  a  line  from  c  parallel  to  C  D 
to  intersect  a  b  produced  for  d. 

26.    Elevations.      Differences 
in    elevation     as    obtained    with    the 
j.  plane  table  may  be  good  enough  for 
f}rt    *  some  topographical  surveys.      They 

may  be  obtained  in  several  ways. 
Except  those  of  the  rudest  character,  they  require  a 
telescope  on  the  alidade,  a  vertical  circle  or  arc,  a 
level  on  the  telescope,  or  both.  The  telescope  level 
should  be  a  striding  level,  of  the  Berger  pattern, 
standing  on  collars  turned  on  the  body  tube  of  the 
telescope.  These  levels  are  readily  attached  and 
detached,  and  cannot  drop  off.  The  telescope  may 
be  furnished  with  stadia  wires  and  the  stadia  used  to 
get  heights  and  distances  both.  The  distances  may  be 
used  to  check  those  obtained  by  platting  on  the  sheet. 
27-  If  a  telescope  is  used,  the  height  of  the 
transit  axis  from  the  under  side  of  the  rule  should  be 
measured  and  kept  in  mind  in  taking  heights.  This 
distance  added  to  the  height  of  the  table  above  the 
ground,  at  any  station  will  give  the  height  of  the  line 
of  sight  of  the  telescope  above  the  ground  when  hor- 


—  23  — 

izontal,  '  or  the   height  of  the    transit   axis    from   the 
ground. 

28.  If  compass  sights  are  used  and  rough  levels 
are  wanted  let  a  line  be  drawn  parallel  to    the   under 
side  of  the  rule   near  the  top  of  the    sights    on    one 
side  for    depressions,   and    near    the    bottom  of    the 
sights  on  the  other  side,   for  elevations.      On  the  rear 
sight  place  a  peep  sight  on  each  of  these   lines.      On 
the    front  sight  mark  graduations,  or  a  scale    of  feet 
per   hundred,    like    the    tangent  scales  on    compass 
sights,   for  elevations  on  one  side  and  depressions  on 
the  other.      Find  the    height  of  each  level  line  from 
the  under  side  of  the  rule  to  be  added  to  the  height 
of  the  table  from  the  ground,   to  get    the    height    of 
the  level  line  of  sight  from  the  ground. 

29.  Taking  differences  of  elevation.    With 
the  plane  table  set  up  at  a  station,  to  get  the   differ- 
ence in   elevation   from  the  ground  under  it   to  the 
ground  at  some  other  station,  or  place.      Place   the 
alidade    central  on  the    table.      Release    the    spindle 
clamp.      Level  the  table  carefully. 

If  the  difference  is  small,  it  may  be  taken  as 
levels  are  taken,  for  short  distances. 

Measure  the  height  of  the  table  from  the  ground 
and  add  this  to  the  height  of  the  level  line  of  sight 
from  the  under  side  of  the  rule  to  get  the  height  of 
the  line  of  sight  from  the  ground.  Call  this  H.  I.  It 
is  plus.  Have  a  rod  held  at, the  place  where  the  dif- 
ference in  elevation  is  to  be  taken.  Sight  by  the  level 
line  of  sight  to  the  rod,  and  take  a  rod  reading.  Call 
this— S.  It  is  minus.  If  compass  sights  are  used, 
hold  a  pencil  point  at  the  forward  mark,  and  be  sure 
the  table  is  well  leveled.  Subtract  the  rod  reading 


—  24  — 

from  H  I  (the  height  of  the  instrument)  to  get  the 
difference  in  elevation.  If  the  result  is  plus,  the 
place  where  the  rod  was  held  is  the  higher;  if  minus, 
it  is  lower.  If  a  telescope  is  used,  it  is  set  level  by  its 
striding  level  and  the  rod  reading  taken  even  if  the 
table  does  not  stand  quite  level.  If  better  results  are 
sought,  have  the  table  well  leveled.  If  much  nicety 
is  required,  lift  the  telescope  from  its  transit  axis 
bearings,  turn  it  over,  replace  it  with  the  transit  axis 
reversed  and  the  telescope  the  other  side  up,  level  it 
again,  and  take  a  second  rod  reading.  Take  the  half 
sum  of  the  two  rod  readings  for  the  correct  one.  It 
will  be  easier  to  keep  the  telescope,  its  line  of  sight, 
and  its  striding  level,  in  good  adjustment  where  much 
of  this  is  to  be  done,  than  it  will  to  take  double 
rod  readings. 

30.  If  the  differences  in  elevation  are  too  large 
to  be  reached  in  the  way  described,  proceed  as  fol- 
lows : 

Get  H.  I.  as  before.  Sight  to  a  mark  on  the  rod 
held  up  at  ithe  place,  the  same  distance  from  the 
ground  the  line  of  sight  is.  With  compass  sights, 
hold  a  pencil  point  to  the  forward  sight  so  it  is  in 
range  from  the  peep  sight  to  the  mark  on  the  rod. 
Take  the  reading  of  the  scale  on  the  forward  com- 
pass sight  at  the  pencil  point.  Multiply  this  by  the 
distance  in  hundreds  of  feet  to  the  rod  for  the  dif- 
ference in  elevation.  Record  the  scale  reading  ± 
with  a  note  of  the  place  where  the  rod  was  held,  and 
do  the  multiplying  at  a  convenient  time.  With  a 
telescope,  set  the  line  of  sight  on  a  mark  on  the 
rod  held  at  the  place  at  the  same  height  from  the 
ground,  the  line  of  sight  is.  Read  the  vertical  circle. 


—  25  — 

Record  the  reading  ±  with  a  note  of  the  place  the 
rod  is  held  at.  Set  the  telescope  level,  by  means  of 
its  level.  Read  the  vertical  circle.  Record  the  read- 
ing HZ  with  the  former  reading.  The  difference  be- 
tween these  two  readings  will  be  the  angle  of  eleva- 
tion, -f  ,  or  depression,  —  ,  of  a  line  joining  the  sur- 
face of  the  ground  at  the  station  where  the  table 
stands  with  the  surface  of  the  ground  at  the  place 
where  the  rod  is  held.  The  distance  between  the 
two  places  may  be  taken  from  the  plane  table  sheet,  or 
measured.  The  difference  in  elevation  equals  this  dis- 
tance multiplied  by  the  tangent  of  the  angle.  For 
greater  refinements,  the  telescope  may  be  turned  over, 
and  the  observations  repeated,  taking  'the  half  sum  of 
the  two  resulting  angles  for  use.  If  the  angle  of  ele- 
vation is  not  over  5°  or  300',  the  difference  in  eleva- 
tion may  be  computed  by  the  following  equation  : 

Biff,  in  El.  =d|  3  A  "  Vl°  A  1^A  ~  V300°  A  +   .00000002d| 


d  =  distance  in  feet.  A  =  the  angle  in  minutes.  If 
A  is  an  angle  of  elevation,  the  result  will  be  plus,  if  an 
angle  of  depression,  the  result  will  be  minus.  The 
last  term  covers  the  correction  for  curvature  and  re- 
fraction and  is  always  plus,  as  shown.  The  terms  in 
the  numerator  give  the  tan.  A  at  5°,  correct  to  6 
places.  For  A  <  5°  tan.  A  will  be  somewhat 
large,  but  only  .00000043  f°r  A  =0°  01'.  This 
equation  can  be  used  within  the  limits  named,  in 
place  of  tables. 

31.  When  a  stadia  is  used,  the  telescope  must 
have  cross  wires  in  it  to  give  stadia  readings,  and  a 
stadia  made  to  fit  the  cross  wires  and  telescope. 
With  this  outfit,  take  and  record  the  height  of  the 


-26  — 

transit  axis  of  the  telescope  from  the  ground  at  the 
instrument  station.  Hold  up  the  stadia  at  the  place 
whose  distance  and  elevation  is  sought.  Set  the  mid- 
dle horizontal  cross  wire  on  a  reading  of  the  stadia, 
equal  to  the  height  of  the  transit  axis  from 
the  ground.  Read  the  vertical  circle.  Record  the 
reading,  ±.  Read  the  stadia  at  both  the  other 
horizontal  cross  wires.  Record  these  readings. 
Set  the  telescope  level,  by  means  of  its  level. 
Read  the  vertical  circle.  Record  this  reading,  ±, 
with  the  former  reading.  For  greater  refine- 
ments, the  telescope  may  be  turned  over,  and 
the  observations  duplicated,  taking  the  half  sum  of 
all  readings  for  their  final  values,  for  use,  with  a 
note  of  the  place  the  stadia  was  held  at.  The  stadia 
notes  may  be  reduced  by  any  suitable  method 
amongst  those  "in  use.  For  the  determination  of  the 
constants  relating  to  the  particular  telescope  in  use 
the  methods  given  in  Searle's  Field  Engineering  22$d 
section,  are  good.  The  stadia  may  be  used  to  de- 
termine contours,  also,  by  the  methods  designated  in 
such  books  as  Johnson's  Surveying,  for  such  work. 
32.  Running  a  Straight  Line. — If  the  line  is 
already  platted,  set  up  the  plane  table  at  a  station  on 
the  line,  and  orient  it.  The  alidade  being  at  the  cen- 
ter of  the  sheet,  mark  its  position  on  the  paper  by 
drawing  fine  lines  at  the  ends  of  the  work  edge  of 
the  rule,  and  at  the  ends  of  the  rule,  so  that  the 
slightest  movement  of  the  rule  on  the  sheet  can  be 
readily  detected.  Be  particular  to  see  that  the  levels 
read  level.  The  line  may  then  be  run,  as  with  a 
transit,  reversing  the  plane  table  on  its  spindle  as  a 
transit  is  reversed,  leaving  the  alidade  undisturbed  on 


—  27  — 

the  sheet.      Extreme   care  is  required  to  keep  from 
moving  the  alidade  on  the  sheet. 

In  case  the  line  is  not  platted  and  it  is  to  be 
prolonged,  set  the  line  of  sight  on  a  mark,  at  a  sta- 
tion of  the  line,  in  the  direction  from  the  plane  table, 
opposite  to  that  in  which  the  line  is  to  go  ;  using  the 
hands  to  move  the  alidade  over  the  sheet  ;  not  dis- 
turbing -the  board,  as  it  is  oriented  ;  and  keeping  the 
edge  of  the  rule  at  the  center  of  the  sheet.  Reverse 
the  telescope  on  its  axis  and  locate  a  line  staff  in  ap- 
parent line  at  the  place  where  the  next  station  is 
needed.  Reverse  the  telescope  on  its  axis,  and  see 
if  the  line  of  sight  strikes  the  back  sight  mark.  If 
it  does  not,  repeat  the  pointings,  correcting  the 
leveling,  or  setting  of  the  plane  table  till  it  will.  Do 
not  disturb  the  orientation  of  the  table  if  the  set- 
ting, or  leveling,  requires  correction,  or  if  it  is  dis- 
turbed, orient  the  table  again.  Having  located  the 
staff  in  apparent  line,  drive  a  stout  stake,  called  a 
plug,  in  its  place.  While  the  plug  is  being  driven, 
try  the  back  sight  again,  see  that  the  table  is  level, 
and  be  ready  to  give  line  for  a  point  on  the  plug. 
Set  the  staff  in  line,  on  the  plug,  with  the  same 
tests  by  back  sighting,  as  before.  Mark  the  point, 
thus  found  on  the  plug,  by  a  temporary  mark,  such 
as  may  be  made  by  sticking  the  point  of  the  line  staff 
into  the  top  of  the  plug  a  little.  Release  the  spindle 
clamp.  Reverse  the  plane  table  on  its  spindle. 
Level  it  again  if  needed.  Set  the  line  of  sight  on  the 
back  sight  mark  by  means  of  the  spindle  clamp  and 
slow  motion  screw,  the  alidade  not  being  moved  on 
the  sheet.  The  table  will  no  longer  be  oriented,  be- 
ing reversed,  and  may  not  be  precisely  in  a  reversed 


—  28  — 

position,  that  is,  not  just  180°  from  its  first  position, 
unless  all  parts  of  the  alidade  are  in  adjustment.  Re- 
verse the  telescope  on  its  axis  and  locate  a  second 
temporary  mark  on  top  of  the  plug,  with  the  same 
tests,  by  back  sighting  as  before.  Drive  a  nail  equi- 
distant from  the  two  temporary  marks  to  mark  the 
station  on  the  line. 

Set  up  a  picket  a  little  behind  the  plane  table,  and 
on  line  for  a  mark  to  back  sight  to  from  the  forward 
station.  A  picket  is  a  stick  3  or  4  feet  long,  about  an 
inch  or  so  thick,  having  a  foot  or  so  of  its  top  straight 
and  blazed  off  white,  with  its  bottom  end  sharpened  for 
driving  into  the  ground.  The  picket  is  set  so  as  to 
stand  firm,  with  its  white  top  end  plumb  and  in  line. 
If  close  to  the  table,  it  may  be  lined  by  looking  back 
over  the  telescope  tube,  or,  if  the  telescope  is  invert- 
ing, by  looking  through  it  from  the  object  end.  If  a 
little  distance  away,  the  picket  may  be  lined  by 
sighting  through  the  telescope  in  the  ordinary  man- 
ner. 

Move  the  plane  table  to  the  forward  station.  Set 
it  up  and  orient  it.  Use  the  station  where  it  stood 
for  a  back  sight,  and  continue  the  line  as  before. 

Some  telescopes  do  not  reverse  on  their  axes. 
They  may  be  made  in  such  cases  to  be  lifted  readily 
from  the  wyes  or  standards.  With  such  a  telescope 
the  above  directions  need  modification.  Open  the 
wye  clips  so  the  telescope  can  be  lifted  out  before 
beginning  to  make  the  pointings.  To  reverse 
the  telescope,  lift  it  from  the  wyes  ;  turn  it  over,  end 
for  end  ;  and  replace  it  with  each  end  of  its  axis  in 
the  same  wye  it  was  in  before.  Otherwise  than  this, 
the  pointings  are  made  the  same  as  with  a  telescope 


that  reverses  on  its  axis.  The  chances  of  disturbing 
the  alidade  on  the  sheet  are  greater  with  this  form  of 
mounting,  and  corresponding  care  and  pains  are  re- 
quired. 

If  the  telescope  does  not  reverse,  and  does  not 
lift  out  of  the  wyes,  run  the  line  by  foresights. 

To  run  a  line  by  foresights,  set  up  the  plane  ta- 
ble at  a  station  of  the  line  and  orient  it.  Mark  a  sta- 
tion in  the  direction  in  which  the  line  is  to  go  to 
sight  to,  to  get  the  line  called  a  foresight.  Set  the 
line  of  sight  on  the  foresight  mark  by  moving  the 
alidade  over  the  sheet  with  the  hands  ;  not  disturb- 
ing the  board,  as  it  is  oriented  ;  and  keeping  the 
edge  of  the  rule  at  the  center  of  the  sheet.  Mark 
the  position  of  the  rule  on  the  sheet.  Locate  a 
point  on  a  plug  beyond  the  foresight,  and  where  it 
can  be  seen  from  the  foresight  station.  See  if  the  line 
of  sight  still  strikes  the  foresight.  If  not,  repeat  these 
operations  till  it  will.  Release  the  spindle  clamp. 
Reverse  the  plane  table  on  its  spindle.  Reverse  the 
alidade  on  the  sheet  by  the  marks  showing  where  the 
rule  first  rested.  Level  the  plane  table  again  if 
needed.  Set  the  line  of  sight  on  the  foresight  by 
means  of  the  spindle  clamp  and  slow  motion  screw. 
The  plane  table  is  no  longer  oriented.  Locate  a  sec- 
ond point  on  the  plug  beside  the  first,  with  the  same 
tests.  Drive  a  nail  equidistant  from  the  two  marks 
on  the  plug  to  mark  the  station  on  the  line. 

Move  the  plane  table  to  the  foresight  station. 
Set  it  up  and  orient  it.  Use  the  new  station  for  a 
foresight,  and  continue  the  line  as  before. 

When  the  line  is  not  platted,  or  it  is  not  neces- 
sary to  orient  the  plane  table  to  do  other  work,  the 


—  30  — 

line  may  be  prolonged  by  placing  the  alidade  to  the 
center  of  the  sheet  without  orienting  the  table,  and 
the  first  pointing  made  with  the  spindle  clamp  and 
slow  motion  screw. 

The  plat  of  the  line  as  prolonged  is  found  on  the 
sheet  by  carefully  continuing  the  piece  that  is  there, 
platting  the  new  stations  located,  and  testing  their  po- 
sition on  the  sheet  by  shots  at  well  determined  points, 
previously  platted. 

33.  Intersecting  Two  Straight  Lines.— 
Extend  the  lines  on  the  ground  and  set  two  plugs, 
with  nails  to  mark  one  of  the  lines,  the  plugs  stand- 
ing one  on  each  side  of  the  other  line,  as  nearly  as 
can  be  judged.  Stretch  a  string  between  the  nails. 
Set  a  third  plug  under  the  string  and  on  the  other 
line.  Drive  a  nail  in  this  plug  where  the  other  line 
crosses  the  string.  This  nail  will  mark  the  intersec- 
tion of  the  lines. 

The  two  first  plugs  may,  by  miss-judgment,  be 
on  the  same  side  of  the  second  line.  In  that  case 
extend  the  string  to  cross  the  second  line  and  tie  it  to 
a  stake.  Or  the  first  line  may  be  marked  by  another 
plug  which  shall  stand  on  the  other  side  of  the  second 
line. 

Sometimes  the  three  plug  method  is  inconvenient 
because  the  ground  is  so  rough.  Then  set  two  plugs 
on  each  line  so  as  to  straddle  the  other  ;  draw  two 
strings  ;  and  set  a  nail  in  a  fifth  plug  to  mark  the 
intersection. 

Sometimes  the  last  plug  comes  considerably  be- 
low the  string,  or  strings, — too  far  below  for  good 
sighting.  In  such  a  case  hold  a  plumb  line  so  as  to 
look  past  it,  and  the  string  being  used  to  line  by. 


—  31  — 

Make  the  plumb-line  cover  the  string  and  the  sight 
will  be  in  a  vertical  plane  with  the  string.  A  weight 
tied  on  a  string  will  do.  Steady  the  hand  holding 
the  plumb  line  on  a  staff  it  needed. 

Sometimes  strings  cannot  be  stretched  between 
the  line  plugs.  Set  up  the  plane  table  over  the  nail 
in  one  of  the  plugs  ;  set  the  line  of  sight  on  the  nail 
in  the  other  and  do  the  sighting  with  the  line  of  sight 
of  the  alidade.  This  takes  more  time.  This  plan 
may  be  used  in  case  the  intersection  plug  comes  in  a 
hole. 

It  may  be  sufficient  for  the  purposes  of  the  work 
to  plat  the  lines  on  the  paper  and  produce  them  there 
till  they  meet. 

3i.  Measuring  the  Angle  Between  the 
Two  Lines.  —  Intersect  the  lines.  Measure  the 
smaller  angle  between  them. 

Set  up  the  plane  table  at  the  intersection  of  the 
lines.  Measure  off  on  the  sides  of  the  smaller  angle 
between  the  lines  from  their  intersection  two  hundred 
feet  or  more  on  each.  Make  the  measurement  care- 
fully, and  set  a  nail  in  a  plug  on  each  line  to  mark 
the  ends  of  the  measurements,  using  the  alidade  to 
line  the  marks  with.  Measure  the  distance  between 
the  two  nails  as  precisely  as  practicable.  Divide  this 
distance  in  feet  by  twice  the  number  of  feet  measure- 
ment off  on  the  two  lines  from  their  intersection. 
The  quotient  will  be  the  sine  of  one-half  the  angle 
between  the  lines.  Look  up  the  corresponding  angle 
in  the  table  of  natural  sines  and  multiply  it  by  two  to 
get  the  angle  between  the  lines. 

The  same  thing  may  be  done  with  a  scale  on  the 
drawing,  using  the  plat  of  the  lines  thereon.  If  five 


—  32  — 

units  of  some  convenient  scale  are  used  in  laying  off 
the  distances  on  the  two  sides  of  the  angle,  one- 
tenth  of  the  distance  between  the  ends  of  the  distances 
measured  in  the  same  unit  will  be  the  sine  of  half  the 
angle  sought. 

One  of  the  lines  may  need  to  be  prolonged  pas 
their  intersection  to  get  the  smaller  angle.  On  the 
ground  this  will  be  done  with  the  plane  table  as  in 
running  a  straight  line.  On  the  drawing,  do  it  with 
the  rule. 

The  measurements  made  on  the  ground  may  be 
used  at  once  with  the  scale  and  dividers  to  lay  the 
angle  down  on  the  drawing  without  finding  the  value 
of  the  angle. 

The  process  applies  to  the  angle  between  two 
lines  of  a  survey,  or  to  the  deflection  angle  from  one 
to  the  next,  since  these  angles  are  supplements  of 
each  other. 

35.  Laying  out  a  Given  Angle.— Plat  the 
angle  on  the  sheet  of  paper  with  its  vertex  at  the 
center  of  the  sheet.  Draw  the  lines  which  form  the 
sides  of  the  angle  out  sufficiently  to  fit  the  rule  of 
the  alidade  to,  or  mark  pieces  of  them  so  the  outer 
end  of  the  rule  can  be  fitted  to  them,  if  it  is  not  de- 
sirable to  draw  the  whole  line.  Or  a  separate  extra 
piece  of  paper  may  be  fastened  on  the  board  for  this 
work.  If  the  extra  paper  is  used  fasten  it  on  first, 
find  and  mark  the  center  of  the  sheet  on  it,  and  plat 
the  lines  from  that.  Set  up  the  plane  table  at  the 
plug  or  station,  that  is  to  be  the  vertex  of  the  angle. 
Orient  the  table  by  one  of  the  platted  lines.  Set  the 
alidade  to  the  other  line.  Range  out  this  line  by  the 


—  33  — 

line  of  sight  of  the  alidade,  and  mark    it   with    plugs 
and  nails  on  the  ground. 

If  the  given  angle  is  much  over  90°  it  will  be 
better  to  plat  its  supplement  on  the  plane  table  sheet, 
and  produce  one  side  of  the  plotted  angle  to  get  a 
plat  of  the  given  angle.  This  refers  to  the  ordinary 
case  in  the  field  where  there  is  no  protractor.  If  a 
protractor  is  used,  the  angle  can  be  laid  down  on  the 
sheet  at  once  with  that.  Without  a  protractor  use 
the  table  of  natural  tangents  or  natural  sines. 

To  plat  an  angle  with  natural  tangents  measure 
off  from  the  vertex  of  the  angle  on  the  given  or  as- 
sumed side  ten  parts  as  large  as  can  be  conveniently 
used,  as  10  or  20  inches.  Here  draw  a  perpendicular 
to  the  given  side  in  the  direction  the  angle  is  to  lie. 
Take  from  the  table  the  natural  tangent  of  the  angle. 
Move  its  decimal  point  one  place  to  the  right.  Meas- 
ure off  on  the  perpendicular  from  the  given  side  the 
resulting  number  of  the  same  parts,  ten  of  which 
were  used  in  locating  the  perpendicular.  Draw  a  line 
from  the  end  of  this  measurement  on  the  perpendic- 
ular to  the  vertex  of  the  angle  for  the  other  side  of 
the  angle. 

To  plat  an  angle  with  natural  sines  take  out  the 
natural  sine  of  half  the  angle  and  multiply  this  sine 
by  two.  With  a  radius  of  ten  parts  draw  the  arc  of 
a  circle  with  the  vertex  as  a  center  and  cutting  the 
given  side.  With  a  radius  of  a  number  of  the  same 
parts  equal  to  twice  the  sine  of  half  the  given  angle 
with  its  decimal  point  moved  one  place  to  the  right, 
and  the  intersection  of  the  arc  and  the  given  side  as  a 
center,  draw  an  arc  cutting  the  first  arc.  Draw  the 


-34- 

other  side  of  the  required  angle  from  its  vertex  to  the 
intersection  of  the  two  arcs. 

30.  Running  a  Traverse.—  Run  and  plat  the 
successive  lines  of  the  survey  in  the  manner  previ- 
ously described  for  the  various  items  of  elementary 
work,  or  determine  the  vertices  of  the  angles  of  the 
lines  of  the  survey  by  any  of  the  methods  given.  The 
result  will  be  a  plat  of  a  continuous  survey  which  is 
often  called  a  traverse.  Strictly  speaking  the  plane 
table  does  not  run  a  traverse  in  the  sense  of  that  term 
in  transit  work,  but  produces  a  plat  of  the  lines  tra- 
versed or  surveyed.  If  computations  are  to  be  based 
on  a  plane  table  plat,  the  lengths  of  the  sides  should 
be  measured  on  the  ground  with  a  steel  tape  and  the 
plat  checked  by  these  measurements.  The  plat 
should  also  be  to  a  large  scale. 

37.  Obstacles  on  Straight  Lines.—  Obstacles 
are  not  frequently  met  with  on  ordinary    plane  table 
surveys.      The  lines  of  the  survey  are  chosen  so  as  to 
avoid  them.      It  is,  however,    quite  possible  that  they 
may  be  met  with.      If  the    scale    of  the    plat    on    the 
board  is  so  small  that  the   object   to  be  passed  would 
appear  insignificant,  or  the  lines    used  to   pass   it   too 
short  for  setting  the  alidade   to,   use  an  extra  piece  of 
paper,  and  plat  the  work  to  a  sufficiently  large    scale 
while  passing  the  object.      This  will  permit  the  work 
to  be  checked,    as  well  as  to  be  done  with  sufficient 
precision. 

38.  Passing  an  Object  on  a  Straight  Line  by 
Deflections.     Fig.    5.—  Fasten   an    extra   piece   of 
paper  over  the  middle  of  the  sheet.     Find  and  mark 
the  center  of  the  sheet  on    the  extra  piece  of  paper. 
Set  up  the  table  and  orient  it  at  some  point  of   A  B, 


—  35  — 

as  at  B,  where  a  line  B  C  can  be  laid  out  clear  of  ob- 
structions. Start  B  C  so  as  to  pass  the  object. 
Draw  lines  from  nt,  the  center  of  the  sheet,  towards 
C  and  towards  A,  of  suitable  length  to  set  the  alidade 
by.  Set  the  alidade  on  B  C.  Release  the  spindle 
clamp.  Turn  the  table  round  on  its  spindle  and  set 
the  line  of  sight  on  B 
A  by  means  of  the 
spindle  clamp  and  slow 
motion  screw,  leaving 
the  alidade  undisturbed 
on  the  sheet.  Set  the 
alidade  again  upon  B 
C  and  draw  a  second 
line  from  m  towards  C. 
This  is  done  for  the 
purpose  of  drawing  a 
line  from  m  that  shall 
make  an  angle  with  the 
line  originally  drawn 
from  m  towards  A 
equal  to  twice  DEC, 
or  to  lay  down  2  D  B 
C  on  the  sheet.  Run 
B  C  to  C,  measuring 
its  length,  or  till  a  sec- 
ond line  C  D,  run  so  that  C  D  B  =  D  B  C,  will  pass 
the  object.  Move  the  table  to  C.  Set  it  up.  Orient 
it  on  C  B  by  the  second  line  drawn  from  m  towards  C 
when  at  B.  The  line  drawn  from  m  towards  A, 
when  at  B,  will  now  show  the  direction  in  which  to  run 
C  D.  Run  out  C  D  accordingly  and  make  it  equal 
by  measurement  to  C  B,  thus  locating  D.  Release 


—  36  — 

the  spindle  clamp.  Re-orient  the  table  by  the  first 
line  drawn  from  m  towards  C  when  at  B,  so  as  to 
orient  the  table  with  respect  to  A  B.  Set  the  alidade 
on  C  D  and  draw  a  line  from  m  towards  D.  Move 
the  table  to  D.  Set  it  up.  Orient  it  by  the  last  line 
drawn. 

Check —  Locate  E,  supposed  to  be  in  A  B, 
produced.  Place  the  alidade  on  the  line  first  drawn 
from  m  towards  C  when  at  B.  Release  the  spindle 
clamp.  Turn  the  table  round  its  spindle  axis  and  set 
the  line  of  sight  on  E  by  means  of  the  spindle  clamp 
and  slow  motion  screw,  leaving  the  alidade  undis- 
turbed on  the  sheet.  See  if  the  line  drawn  from  m 
towards  A  when  at  B  now  points  to  C.  If  it  does 
not,  review  the  work  and  correct  the  errors. 

If  it  is  considered  to  be  sufficiently  accurate,  the 
point  C  may  be  determined  by  resection  from  A  B, 
and  B  C  found  from  the  plat.  Good  practice  would 
make  use  of  C  in  determining  other  points  and  ob- 
jects required  by  the  survey.  In  such  a  case  B  C  and 
C  D  would  be  platted  on  the  sheet  with  other  sta- 
tions. The  distance  B  D  may  be  found  by  platting 
B  C  and  C  D  to  a  large  scale  from  the  center  of  the 
sheet  on  the  extra  sheet  of  paper  and  measuring  the 
.distance  B  D  between  their  ends  to  the  same  scale. 
Or  measure  the  angle  B  C  D  on  the  ground  with  the 
steel  tape  or  on  the  plat  with  the  scale  (34)  and  find 
B  D  =  2  B  C  sin  ^  BCD. 

For  a  check  upon  such  work  run  past  the  object 
more  than  once,  locating  C  in  a  different  place,  or 
prolong  B  C,  and  set  another  point  on  line  beyond  D. 
Run  past  on  opposite  sides  where  practicable.  Set 
up  the  table  at  any  one  of  the  points  beyond  the  ob- 


H 


—  37  — 

ject.  Orient  it  by  the  line  to  the  station  where  the 
table  stands.  Locate  E  as  before.  See  if  the  other 
points  beyond  the  object  are  in  the  line  to  E.  Ap- 
ply the  check  above  given  at  any  of  the  points  be- 
yond the  object  and  see  if  it  fits. 

39.  Passing  an    Object    on    a 
Straight  Line  by  Traversing.    Fig. 

6. —  Run  a  traverse  past  the  object, 
platting  the  lines  of  the  traverse.  The 
stations  on  the  traverse  may  be  deter- 
mined by  measurement,  resection,  or  any 
of  the  usual  methods.  Extend  the  plat 
of  the  original  line  on  the  sheet  to  the 
forward  part  of  the  plat  of  the  traverse. 
Draw  a  line  on  the  plat  from  the  last 
station  of  the  traverse  so  as  to  intersect 
the  straight  line  produced  past  the  ob- 
ject, and  in  a  position  to  be  conveniently 
run  out  on  the  ground  to  reach  the 
original  line  at  a  point  beyond  the  object. 
Measure  the  length  of  this  line  on  the 
sheet  with  the  scale.  Having  set  up 
the  plane  table  at  the  last  station  of  the 
traverse  on  the  ground  and  oriented  it, 
run  out  the  above  described  line,  meas- 
ure out  its  length  and  locate  a  station 
beyond  the  object  on  the  original  line 
produced.  Continue  the  line  by  means 
of  the  plat  of  it  on  the  sheet. 

The  above  method  is  comparatively  rough,  but 
may  be  used  in  some  cases.  It  is  without  a  check. 
To  improve  the  results,  the  traverse  may  be  platted 


C 


—  38  — 

on  an  extra  sheet  to  a  large  scale,  and  the  closing 
distance  more  carefully  measured.  More  than  one 
traverse  may  be  carried  past  the  object  and  the  re- 
sults compared.  All  the  stations  on  all  the  traverses 
riiay  be  useful  in  extending  or  perfecting  the  sur- 
vey. The  points  beyond  the  object  may  be 
checked  by  resection  on  other  platted  stations. 

The  traverse  may  be  carried  a  sufficient  dis- 
tance beyond  the  object,  as  to  D,  E,  and  F,  to 
permit  of  lines  being  drawn  from  E,  and  F,  to  H, 
the  point  where  the  closing  line  D  H  intersects  A  B, 
produced  past  the  object.  Locate  H  by  intersections 
from  D  and  E,  and  check  it  by  resection  on  F.  A 
second  point  K  may  be  located  by  similar  means.  A 
number  of  these  stations  may  be  used  to  check  the 
locations  of  one  another,  as  well  as  form  the  basis  for 
extending  the  survey. 

40.  Passing  an  Object  on  a  Straight  Line 
With  the  Plane  Table  Alone.  Fig.  7:— Set  up 
the  plane  table  at  B.  Orient  it  by  B  A.  Run  out 
two  lines  B  C  and  B  D,  making  such  equal  angles 
with  A  B  that  they  will  pass  the  object.  Set  a  station 
D  on  B  D,  so  that  D  C  perpendicular  to  A  B  will 
pass  the  object.  Set  plugs  as  near  C  as  can  be  es- 
timated preparatory  to  intersecting  D  C  with  B  C. 
Move  the  table  to  D.  Set  it  up.  Orient  it  by  D  B. 
Assume  a  point  d  for  the  plat  of  D,  on  the  plat  of  D 
B,  so  as  to  give  a  good  length  to  the  lines  of  the  dia- 
gram being  developed  on  the  sheet  to  set  the  rule 
of  the  alidade  by.  From  d  lay  out  a  line  on  the 
sheet  to  represent  the  plat  of  D  C,  perpendicular  to  a  b . 
Run  this  line  out  on  the  ground  and  intersect  it  with 
B  C  at  C.  Continue  the  line  on  the  plat  and  find  c. 


—  3D  — 

Start  the  lines  D  E  and  D  F  for  locating  E  and  F  at 
convenient  angles  from  D 

B,  and  draw  d  e  and  dfon 
the    plat  to   intersect  a  b 
produced.      Set  plugs  on 
D  E  and  D  F,  as  near  E 
and  F  as  can  be  estimated 
preparatory    to  intersect- 
ing these   lines  with  C  E 
and    C   F.      Lay  out  c  e 
and  c  f  from  c  on  the  plat, 
making  B  C  E  =  B  D  E   J) 
and    B    C    F  — B    D    F. 
Move   the  plane  table  to 

C.  Set    it     up     at    C. 
Orient  it  by  C  B.      Test 
its     position    by     C     D. 
Run  out  C   E  and   C   F. 
Intersect  them  with  D  E 
and  D  F,   locating  E  and 

F.  Move  the  plane  table  and  set  it  up  at  E  or  F.  as 
may  be  most  desirable.  Orient  it  by  the  line  to  C. 
Test  its  position  by  F  E  and  the  line  to  D.  Con- 
tinue the  line  to  H,  or  any  point  beyond,  using  F  E 
( »r  A  B,  as  they  should  be  the  same. 

This  work  may  require  an  extra  sheet  of  paper, 
but  the  practitioner  will  readily  decide  such  matters 
for  himself.  C,  D,  E,  and  F,  may  be  used  as  regu- 
lar stations  on  the  survey.  The  object  may  be  plat- 
ted as  this  work  proceeds. 

41.  Passing  an  Object  on  a  Straight  Line 
by  Resection.  Fig.  8.—  Locate  E  in  A  B  pro- 
duced beyond  the  object.  Choose  D  as  anv  con- 


40  — 


venient  station  on  the  same  side  of  the  object  that  E 
is.  Set  up  the  table  at  E  and  orient  it  by  A  B.  Set 
the  alidade  on  E  D,  and 
draw  a  line  toward  D,  any- 
where on  the  sheet.  Move 
the  table  to  D.  Set  it  up. 
Orient  it  by  the  last-named 
line.  Resect  on  B  and  C 
to  get  D.  Set  the  alidade 
on  D  E.  Draw  a  line  from 
d,  parallel  to  D  E  to  inter- 
sect a  b  produced  to  get  e. 
Measure  c  e  on  the  plat. 
As  a  check,  measure  D  E 
on  the  ground  and  d  e  on  the 
plat.  See  if  they  agree. 

The  larger  the  scale 
used,  the  more  perfectly  will 
the  result  be  accomplished. 

42.  Inaccessible   Lines.— Problems  relating 
to  inaccessible    horizontal    distances    may    be    solved 
with  the  plane  table  without  calculation.      This    is  in- 
dicated by  Figs.  6.  7,  and  8.      The    relative  elevation 
of   the  points   between  which  the  required  distance  is 
found  in  these  cases  has  theoretically   nothing   to    do 
with  the  application  of  these  principles.      The  same  is 
true  of  Fig.  3.      A  detailed   account    of   problems  re- 
lating to  inaccessible  lines  is  not    given,    because    the 
methods  illustrated  by  these  figures  may  be    adapted 
to  all  the  ordinary  cases  to  be    met    with.      Problems 
of  this  class  are  rare  in  plane  table  work. 

43.  Running  Circular  Curves.  Fig.  9.—  In- 
tersect the  tangents  A  I  and  B  I  at  I  (33).   Set  up  the 


—  41  — 

plane  table  at  I.  Measure  the  angle  D  I  B=  A  (34)- 
Choose  a  suitable  curve.  The  curve  may  be  desig- 
nated by  its  degree,  like  a  railroad  curve  when  the 
station  chords,  as  Pt 
P2,  P2  P3,etc.,  are  100 
feet.  When  the  station 
chords  are  not  100  feet, 
the  curve  may  be 
named  by  using  the  sta- 
tion chord  for  the  nu- 
merator of  a  fraction 
and  the  angle  the  sta- 
tion chord  covers  at  the 
center  of  the  curve,/? 
as  the  denominator,  f 
thus 


20 


which  signifies  that  a 
chord  of  20  feet  covers 
an  angle  at  the  center 
of  the  curve  of  10°. 
This  device  may  be 
called  the  ' l  Name' '  of 
the  curve.  It  corres- 
ponds to  the  conception  used  in  the  term  ' l  Degree' ' 
used  in  railroad  curves,  which  means  the  angle  at 
the  center  covered  by  a  station  chord  of  100  feet,  or 
the  numerator  of  the  above  fraction  becomes  100. 
If  the  curve  is  designated  by  its  degree,  take  its  radius 
R,  with  log.  R,  from  a  curve  table.  If  the  curve 
is  "named,"  as  above,  compute 


L 


—  42  — 


R=        —  ,    where  c  =  the 
sin  ^2  d 

station  chord  and  d  the  station  angle  it  covers.  Com- 
pute the  tangents  I  A  =  I  B  =  T=R  tan  ^  A,  and 
the  vertex  distance  I  K  =  V  =  T  tan  %  A.  Set  the 
alidade  on  I  M.  Measure  from  I  towards  M,  I  B  — 
T,  and  locate  B.  Set  a  plug  or  stake  and  nail  at  B. 
Bisect  the  angle  a  i  b  on  the  sheet,  denning  the  di- 
rection of  i  k.  The  plane  table  should  be  oriented 
with  respect  to  I  A  and  I  B,  using  i  a  or  i  b  to  orient 
the  table  by,  and  check  its  position  by  the  other. 
Set  the  alidade  by  i  k.  Range  out  the  line  I  K  on 
the  ground.  Measure  from  I  to  K,  I  K  =  V,  and 
locate  K.  Set  a  stake  and  nail  at  K.  Set  the  ali- 
dade on  I  L.  Measure  from  I  towards  L,  I  A  =  T, 
and  locate  A.  Set  a  plug,  or  stake,  and  nail  at  A. 
Move  the  plane  table  to  A.  Put  on  an  extra  sheet  of 
paper,  if  not  already  done  at  I.  Set  up  the  table  at 
A.  Measure  from  the  last  preceding  station,  as  L,  to 
A,  as  L  A.  Mark  a  witness  stake  for  A,  bearing  the 
line  mark,  as  Q  ;  the  number  of  the  regular  station 
at  L,  plus  the  distance  from  L  to  A,  as  147  -f  62.37  ; 
the  letters  P.  C.  ,  point  of  curve  ;  the  degree  or 
name  of  the  curve  as  4°  or 


20 

20 c 


the  direction  of  the  curve,  as  L.   for    left,   or  R.  for 
right  ;  and   the  angle  D  I  B  =  A,  as  70°  34'  30'  ', 


-43  — 
to  read  from  the  top  of  the  stake  downwards,  —  as 

Q 

147  +  62.37 

P.  C. 

20' 

^0~° 

C.  R. 

C.  R.  for  70°  34'  30".  Drive  this  stake  about  a 
foot  to  the  right  of  A,  with  its  station  number  fac- 
ing L. 

Subtract  the  distance  L  A  from  the  regular  sta- 
tion distance,  Pl  P2  to  get  the  first  short  chord,  as  A 
Pl  =clm  Find  one-half  the  angle  this  chord  covers 
at  the  center  of  the  curve,  by 


sin 


R 


Subtract  %  d^  from  ^  A-  Divide  the  result  by  ^  d. 
The  remainder  found  by  this  division  will  be  ^  df2, 
one-half  the  central  angle,  covered  by  the  short 
chord  P3  K  =  r2.  Compute  r2  by 


<:2  =  2  R  sin   ^  d 


Subtract  ^  dl  from  ^  LJ.  Divide  the  result  by  ^  d. 
The  remainder  found  by  this  division  will  be  ^  ^3, 
one-half  the  central  angle  covered  by  the  short  clos- 
ing chord  P.  B  —  £8.  Compute  c^  '  y 

<:3  =  2  R  sin  ^  ^3. 

Plat  the  angle    I  A  Pt  =  ^  ^!   with  its  vertex 
at  the  center  of  the  sheet  and  with  long  sides  for   set- 


—  44  — 

ting  the  alidade  by  in  running  the  curve.  On  the 
same  side  of  this  angle,  supposing  the  observer  to  be 
at  its  vertex,  that  the  curve  is  to  run.  plat  ^  d,  one- 
half  the  regular  station  angle.  Number  these  lines 

0.  i,  2,  in  the  order  they  come  in,  laying  down    the 
angles  platted.     Still  on  the  same  side  of  the  last  an- 
gle platted,  or  continuing  from  line   2,  plat  ^  d^  and 
number  this  line  3.      From    line  2    also  plat  ^  d^  in 
the  same  direction  and  number  this  final  line  4. 

Set  the  alidade  on  o.  Set  the  line  of  sight  on 
A  I  by  means  of  the  spindle  clamp  and  slow  motion 
screw.  Set  the  alidade  on  i.  Take  a  distance  on 
the  chain,  or  tape,  equal  to  c  1  =  A  Pr  Hold  one 
end  of  this  distance  at  A.  Swing  the  other  end  of 
this  distance  on  the  tape  into  line  at  Pl.  Drive  a 
stake  at  Plt  marked  for  the  number  of  that  station, 
with  the  number  facing  A.  Test  this  stake  for  line 
and  distance,  and  when  it  is  adjusted  for  both,  drive  a 
nail  in  it  to  mark  the  station  Pj.  Set  the  alidade  on 
2.  Hold  one  end  of  a  distance  on  the  tape,  equal  to 
the  regular  station  chord,  at  P1 .  Swing  the  other  end 
into  line  at  Pj.  Set  a  stake  at  P2  marked  for  that 
station,  with  the  same  tests  that  were  used  in  setting 
Pj.  Set  the  alidade  back  on  line  o.  See  if  the  line 
of  sight  still  ranges  the  tangent  -A  I.  If  not,  review 
the  work  and  correct  the  errors.  Set  the  alidade  on 

1.  By  means  of  the  spindle  clamp  and  slow  motion 
screw  set  the  line  of  sight  on  P2.     Set  the  alidade  on 

2.  Measure  a  regular  station   chord  from  P2  and  lo- 
cate P3  as  P2  was  located.      Set  the  alidade  again  on 
i,  and  see  if  the   line   of  sight  strikes  P2.      If    not, 
review  the  work  from    P2,     and  correct  the   errors. 
The  alidade  being  set  on  i ,  set  the  line  of  sight    on 


—  45  — 

P3  and  continue  the  curve  as  before,  making  the 
various  checks  and  tests  to  insure  trustworthy  work. 
So  continue  the  work  till  the  station  next  preceding 
K,  the  middle  of  the  curve,  is  reached.  This  is  P3  in 
Fig.  9.  After  P3  is  set,  and  before  shifting  the  table 
on  its  spindle  preparatory  to  locating  the  next  regular 
station,  set  the  alidade  on  line  3.  See  if  the  line  of 
sight  strikes  K.  Measure  the  distance  from  P3  to  K. 
See  if  this  equals  <r2.  If  either  of  these  checks  fail, 
review  the  work  and  correct  the  errors.  Continue 
the  curve  from  P3,  not  from  K,  till  the  station,  as  P5, 
next  preceding  the  P.  T.  at  B  is  reached.  After  P5  is 
set  and  checked,  set  the  alidade  on  line  4.  See  if  the 
line  of  sight  strikes  the  P.  T.  at  B.  Measure  from  P5  to 
B.  See  if  this  distance  is  ^3.  If  either  of  these 
checks  fail,  review  the  work  and  correct  the  errors. 
Mark  a  witness  stake  for  the  P.  T.  with  the  line 
mark  ;  the  station  number  of  P5  plus  the  distance 
P5  to  B  ;  P.  T.  ;  the  name  of  the  curve  ;  L  or  R  ; 
and  the  value  of  A-  Drive  this  stake  about  a  foot 
to  the  right  of  B,  with  the  number  facing  P..  Sub- 
tract c^  from  the  regular  station  distance  to  get  the 
distance  from  B  to  M,  the  next  regular  station  be- 
yond B. 

With  the  plane  table  oriented  so  the  line  4  is  di- 
rected to  B,  set  the  alidade  on  A  I,  and  draw  a  line 
from  m  by  the  edge  of  the  rule.  Move  the  table  to 
B.  Set  it  up  over  B.  Set  the  alidade  on  m  i,  the 
last  line  drawn.  By  means  of  the  spindle  clamp  and 
slow  motion  screw,  set  the  line  of  sight  on  A.  Set 
the  alidade  on  line  4.  See  if  the  line  of  sight  ranges 
on  the  tangent  B  I.  If  not,  review  the  work  and  cor- 
rect the  errors. 


Range  out  the  tangent  I  B.  Measure  c — r3  from 
B  to  M,  and  locate  the  first  regular  station  on  the  tan- 
gent beyond  B,  at  M.  Continue  the  tangent  as  a 
straight  line  (32). 

In  case  the  whole  curve  cannot  be  seen  from  A, 
run  as  far  as  practicable,  as  to  P2  in  Fig.  9.  After 
checking  the  location  of  the  last  station,  set  the  ali- 
dade on  line  i,  and  make  the  line  of  sight  strike  the 
last  station  by  means  of  the  spindle  clamp  and  slow 
motion  screw,  just  as  if  preparing  to  locate  the  next 
regular  station.  Set  the  alidade  on  A  I.  Draw  m  i 
from  m  towards  I  by  the  edge  of  the  rule.  Move  the 
plane  table  to  P2.  Set  it  up  over  P2.  Turn  the  table 
so  the  same  side  of  the  sheet  is  forward  the  way  the 
curve  is  going,  that  was  forward  when  at  A.  Set 
the  alidade  on  i  m,  the  last  line  drawn,  to  look  back 
to  A.  By  means  of  the  spindle  clamp  and  slow  mo- 
tion screw  set  the  line  of  sight  on  A.  Set  the  ali- 
dade on  line  2.  Locate  the  next  regular  station  in 
the  usual  manner,  measuring  a  station  chord  from  P2. 
Check  its  location  by  a  back  sight  on  A.  Continue 
the  curve  as  before  directed  for  the  table  standing  at 
A  ;  also  check  on  K  as  before  directed. 

Should  it  be  necessary  to  move  up  again,  let  P4 
Fig.  9  be  the  station  to  be  used  as  the  next  instru- 
ment station.  Set  the  alidade  on  i  m  again.  By  means 
of  the  spindle  clamp  and  slow  motion  screw,  set  the 
line  of  sight  again  on  A.  Set  the  alidade  on  line  1. 
Make  or  find  a  mark  in  apparent  line  on  the  tangent 
E  F.  Call  this  mark  F.  Release  the  spindle  clamp 
and  set  the  line  of  sight  on  P4  by  means  of  the  spin- 
dle clamp  and  slow  motion  screw,  the  alidade 
being  still  set  on  line  i.  Set  the  alidade  on  P2 


F.  Draw  a  line  m  f  by  the  edge  of  the  rule.  Move 
the  plane  table  and  set  it  up  over  P4.  Turn  the  same 
part  of  the  sheet  to  be  forward  as  before.  Set  the  ali- 
dade on  f  m  to  look  back  to  P2.  Set  the  line  of 
sight  on  P  2  by  means  of  the  spindle  clamp  and  slow 
motion  screw.  Continue  the  curve  as  before. 

Having  moved  up  any  number  of  times  on  a 
curve,  let  P4  be  the  instrument  station  from  which  B 
can  be  seen  and  the  curve  closed.  Before  beginning 
work  from  P4'  after  the  table  is  oriented,  set  the  ali- 
dade on  line  i  and  make  or  find  a  mark  in  appar- 
ent line  on  the  tangent  G  H.  Call  this  point  H. 
Having  closed  the  curve  on  B  by  line  4,  leave  the 
table  with  4  pointing  to  B,  and  set  the  alidade  on  P4 
H.  Draw  m  h,  by  the  edge  of  the  rule.  Move  the 
table  to  B.  Set  it  up  over  B  and  orient  it  by  h  m,  and 
a  back  sight  on  P4.  Set  the  alidade  on  line  4.  See 
if  the  line  of  sight  ranges  the  closing  tangent  B  I.  If 
not,  review  the  work  and  correct  the  errors. 

To  lay  out  a  tangent  at  any  station  as  P4  and  set 
up  the  plane  table  there,  orient  it  preparatory  to 
continuing  the  curve.  Set  the  alidade  on  line  i ,  or 
whatever  line  was  pointing  to  the  instrument  station 
from  the  previous  instrument  station  when  a  point- 
ing was  made  on  the  tangent  there,  preparatory  to 
moving  up.  The  line  of  sight  then  ranges  on  the 
tangent. 

Obstructions  to  running  the  curve  that  cannot 
be  removed  may  often  be  passed  by  a  long  cnord. 
Compute  the  distance  from  the  station  where  the 
plane  table  stands  to  the  station  beyond  the  obstruc- 
tion which  can  be  seen,  by 

C,  =2  R  sin  y2  Dj, 


-48- 

where  Cl  is  the  required  distance,  or  long  chord, 
and  Dj  is  the  central  angle  covered  by  C15  found  by 
the  number  of  station  chords  on  the  curve  between 
the  two  ends  of  Cl.  Run  the  curve  as  far  as  practi- 
cable towards  the  obstruction.  Having  checked  the 
location  of  the  last  station,  and  without  shifting  the 
plane  table,  plat  from  line  2  an  angle  equal  to  one- 
half  the  central  angle  included  between  the  last  sta- 
tion located  and  the  station  at  the  far  end  of  C1. 
Call  the  line  thus  laid  down  5.  Set  the  alidade  on 
2.  See  that  the  table  is  level  and  the  line  of 
sight  strikes  the  last  station  located.  If  need  be, 
level  the  table  and  set  the  line  of  sight  on  the  last 
station  located  by  the  spindle  clamp  and  slow  motion 
screw.  Set  the  alidade  on  5.  Run  this  line  a  dis- 
tance Cl  from  the  station  where  the  table  stands  and 
locate  a  station  beyond  the  obstruction.  Mark  the 
tangent  at  the  instrument  station,  as  directed  in  the 
last  paragraph.  Orient  the  table  so  that  5  will  point 
to  the  station  located  beyond  the  obstruction.  Set 
the  alidade  on  the  tangent  and  draw  a  line  by  the 
edge  of  the  rule  from  the  center  of  the  sheet.  Move 
the  table  to  the  station  located  beyond  the  obstruc- 
tion. Set  it  up  there.  Turn  the  table  so  the  same 
side  is  forward  as  before.  Set  the  alidade  on  the  last 
line  drawn,  to  look  back  at  the  station  the  table  was 
moved  from.  Set  the  line  of  sight  on  that  station 
by  means  of  the  spindle  clamp  and  slow  motion 
screw.  Set  the  alidade  on  5.  Mark,  or  find  a  point 
in  apparent  line  for  a  mark  on  the  tangent  at  the  new 
station.  Set  the  alidade  on  i.  Set  the  line  of  sight 
on  the  tangent  by  the  spindle  clamp  and  slow  motion 
screw.  Set  the  alidade  on  2.  Measure  from  the  in- 


OF  THE 

UNIVERSITY 


strument  station  a  station  chord,  and  loCatC  the  next 
regular  station  on  the  curve.  Continue  the  curve  as 
before. 

To  run  the  curve  backwards  to  the  obstruction, 
set  the  alidade  on  2,  instead  of  i.  Set  the  line  of  sight 
on  the  tangent  by  the  spindle  clamp  and  slow  motion 
screw.  Set  the  alidade  on  i,  so  as  to  look  back  on 
the  curve.  Measure  from  the  instrument  station  a 
station  chord  and  locate  the  next  regular  station 
backwards  on  the  curve.  Continue  the  curve  back- 
ward the  same  as  forward,  except  that  lines  i  and  2 
exchange  places  in  the  order  of  their  being  used. 

Where  a  plus  station  occurs,  one-half  the  partial 
angle  covered  by  the  chord  to  it  may  be  laid  off  on 
either  side  of  one-half  the  regular  angle,  i  m  2,  as 
may  be  required,  as  was  done  for  starting  the  curve. 

Wishing  to  know  the  plus  station  that  would 
come  on  a  designated  line  running  from  the  instru- 
ment station,  measure  the  angle  (34)  between  it  and  a 
line  to  the  last  station  located,  or  between  it  and  the 
tangent  at  the  instrument  station.  Call  this  angle 


Find  n  = 


Reduce  the  decimal  part  of  n  to  feet,  or  the  'units  the 
station  distance  is  measured  in.  Add  the  resulting 
number  and  plus  to  the  number,  and  plus  (if  any),  of 
the  station  where  the  plane  table  stands  to  get  the 
number  and  plus  of  the  station  at  the  end  of  the  line 
if  \  d±  was  measured  from  the  tangent  at  the  instru- 
ment station,  or  the  last  station  located,  if  reckoning 
from  there. 


—  50  — 

Lay  the  angles  out  on  the  sheet  with  great  care. 
Be  especially  careful  in  setting  the  alidade  to  the  lines 
on  the  sheet. 

44.  Laying  Out  Parabolic  Curves.     Fig. 10. 

—  These  curves  may  be  used  where  the  tangents  I  A 
and  I  B  are  equal  or  unequal.  The  curve  cuts  the 
equally  spaced  breadths  i-i  2-2  etc.,  drawn  parallel 
to  the  line  I  M,  joining  I  with  M,  the  middle  point 
of  A  B,  so  as  to  make  the  following  proportions.  The 
distances  3~P3;  5~P5  ;  etc.,  from  the  tangents  I  A 
and  I  B,  and  P2-2  ;  P6-6;  et  .,  from  the  chord  A  B 
are  equal,  if  equally  removed  from  I  M.  P4  is  in 
the  middle  of  I  M.  i-Pj ;  2-P2;  3~P3 ;  etc.,  are  in 
the  ratios  of  the  squares  of  the  natural  numbers.  Pr-i ; 
P2  -2 ;  P3-  3,  etc. ,  are  in  the  ratios  of  the  products  of  the 
natural  numbers  written  from  A  to  M,  the  one  at  M 
repeated,  and  continued  back  to  A,  as  in  the  figure. 
This  may  be  represented  as  follows  : 

0  o          A  A         o  8  o 

1  i         I-P!        P^i      i  7  7 

2  4  2-P2  P2~2       26  12 

3  9         3-P3       p3-3     35         15 

4  16         I-P4       P4-M    44        16 

Make  an  even    number  of   divisions    between  A 

and  I  M. 

Intersect  the  tangents  A  I  and  B  I  at  I  (33). 
Survey  and  plat  these  tangents.  Set  up  the  plane 
table  at  I.  Locate  the  tangent  points  A  and  B  at 
equal  or  unequal  distances  from  I,  as  may  be  desired. 
It  may  be  well  to  vary  the  positions  of  A  or  B  slightly 
in  order  to  make  the  spacings  on  the  tangents,  I~3  ; 
3-2  ;  or  1-5  ;  5-6  ;  etc. ,  of  a  convenient  length  to  lay 


7 


S 


-51- 

out.  Lay  out  and  locate  1,2,  3,  etc.,  on  A  I  ;  and 
5,  6,  7,  etc.,  on  I  B,  so  as  to  divide  each  tangent 
into  the  same  even  number  of  equal  parts.  Move  the 
plane  table  to  A.  Set  it  up  at  A  and  orient  it  by  A 
I.  Set  the  alidade  on  A  B. 
Measure  A  B  and  plat  it.  See  if 
it  checks  on  b.  If  not,  review  the 
work  and  correct  the  errors.  Lo- 
cate M,  the  middle  point  of  A  B. 
Locate  points  i  ;  2  ;  etc.,  on  A 
B,  so  as  to  divide  A  B  into  the 
number  of  equal  parts  in  both 
tangents  A  I  and  I  B.  Measure 

1  M  on   the  ground   and  on  the 
plat.    See  that  they  agree.    Divide 
^  I  M  by  the  square  of  the  num-  •* 
ber  of  divisions  in  the  tangent  A 

I,  to  get  the  unit  measure  for  the  3 
work.  Multiply  this  unit  by  the 
ratio  numbers  formed  as  above 
noted  to  get  the  measurements 
for  locating  points  on  the  curve. 
Figure  out  both  sets.  Add  them/ 
in  succession  for  the  various 
points  P:  ;  P2  ;  etc.  See  if  the 
results  are  in  proportion  to  the 
natural  numbers  i,  2,  3,  etc.  If 
not,  review  the  figures  and  cor- 
rect the  errors.  Stretch^  a  cord,  or  tape,  from  i 
to  i.  Locate  Pl  on  this  line  by  the  measure- 
ments for  Pl .  Stretch  a  cord,  or  the  tape,  between 

2  and  2.      Locate  P2   on    this    line   by    the  measure- 
ments for  P,.      So  continue  for  each    of   the  several- 


—  52  — 

stations.  Plat  them  on  the  sheet  in  the  same  man- 
ner. Measure  from  A  to  P1  ;  P:  to  P2  ;  P2  to  P8  ; 
etc.,  around  the  curve.  Record  these  measure- 
ments. 

To  lay  out  a  tangent  at  any  point,  as  P3.  Di- 
yide  the  number  of  the  point  by  2.  The  quotient 
will  show  the  number  of  spaces  on  A  I,  from  A  to  E, 
the  intersection  of  the  required  tangent  with  A  I  ;  and 
the  number  of  spaces  on  I  B,  from  I  to  F  the  inter- 
section of  the  required  tangent  with  I  B.  Measure  off 
these  spaces,  and  locate  E  and  F.  Run  a  line  from 
E  to  F.  See  if  it  passes  through  P3.  If  not,  re- 
view the  work  and  correct  the  errors. 

Any  number  of  tangents  may  be  laid  out  as  above 
noted,  the  measurements  computed  from  them  to 
points  on  the  curve  ;  and  these  used  in  locating  sta- 
tions ;  either  with  respect  to  A  B,  or  the  chord  join- 
ing the  tangent  points  of  these  tangents  with  each 
other,  or  with  A  or  B.  By  this  means  obstructions 
may  be  passed  or  difficulties  overcome. 

The  station  as  P3,  at  which  a  tangent  as  E  F  is 
laid  out,  divides  that  tangent  into  two  parts  whose 
ratios  are  expressed  by  the  number  of  station  distances 
between  the  station  and  A  and  B.  E  P3  is  j^sths 
and  P3  F  is  ^ths  of  E  F.  This  principle  may  be 
used  to  avoid  obstructions. 

45.  Direction  of  a  Line.  Fig.  11.— The 
direction  of  a  line  may  be  found  by  observing  stars  at 
their  elongations.  The  stars  near  the  pole  are  the 
most  useful  for  this  purpose.  Their  names  and  rel- 
ative positions  can  be  readily  learned  from  a  star  map 
or  a  cheap  planisphere.  This  will  enable  one  to' find 
the  stars  required  for  observation,  or  to  tell  the  stars 


-53- 

that  may  be  observed  at  any  time.  The  apparent 
daily  motions  of  the  stars  are  in  arcs  of  circles.  When 
a  star  is  at  that  part  of  its  daily  motion  path,  where  it 
appears  to  move  vertically,  or  follow  the  range  of  a 
plumb  line,  it  is  at  its  elongation.  At  the  eastern 
elongation  it  is  going  upwards,  at  the  western  down: 
wards.  Western  elongations  are  more  convenient 
for  observation  with  ordinary  instruments.  The  ob- 
servation is  an  easy  one  to  make,  as  the  star  moves 
in  a  nearly  vertical  direction  for  some  time.  The  ob- 
servation consists  in  watching  the  star  near  its  elonga- 
tion, and  keeping  the  range  of  a  line  to  it.  Finally 
get  the  direction  of  a  line  to  the  star  when  at  its  elon- 
gation. This  is  done  by  following  the  star  with  the 
vertical  cross  wire  in  a  telescope,  or  by  moving  a 
compass  sight  on  a  level  board,  even  keeping  the 
star  behind  a  suspended  plumb  line,  when  watching 
it  through  the  compass  sights, —  or  .by  any  similar 
device.  For  further  information  regarding  this  kind 
of  work  consult  an  article  on  the  ' '  Direction  of  a 
Line,  "  in  the  TECHNIC  for  1892. 

Examine  the  plane  table  and  find  how  large  an  an- 
gle of  elevation  can  be  observed  with  the  rule  stand- 
ing on  the  board  when  sighting.  Unless  this  angle 
considerably  exceeds  the  latitude  of  the  place  of  ob- 
servation, a  diagonal  eyepiece  will  be  required  to  the 
telescope.  If  compass  sights  are  used  the  observa- 
tion will  need  to  be  made  on  an  entirely  different 
plan. 

To  illuminate  the  cross-wires,  tie  a  piece  of  un- 
glazed  white  paper  (common  newspaper)  over  the 
objective  by  a  string  around  the  telescope  tube  back 
of  the  lens,  with  a  hole  in  the  paper  centrally  over 


—  54  — 

the  objective.  This  hole  may  be  from  %"  to  y&" 
broad.  Light  shining  upon  the  outside  of  this  paper 
will  make  the  cross-wires  visible.  A  lantern  with  a 
lens  throwing  light  in  one  direction  is  best.  A  com- 
mon lantern  may  be  made  use  of  by  nearly  surround- 
ing it  with  something  to  shut  off  the  light.  The  light 
must  not  shine  in  the  observer's  face.  It  must  there- 
fore shine  obliquely  from  the  side  upon  the  paper 
over  the  objective.  The  light  may  be  toned  to  any 
faintness  within  the  telescope  by  withdrawing  the 
lantern,  or  changing  the  obliquity  of  the  beam  of 
light. 

Choose  some  point  of  the  line  whose  direction  is 
sought,  or  reference  line,  from  which  a  clear  view  of 
the  stars  to  be  observed  may  be  obtained  for  an  in- 
strument point.  Fasten  a  lantern  at  some  other 
point  of  the  reference  line  to  sight  to.  A  common 
lantern  about  a  mile  away  will  do  very  well.  If  nec- 
essary to  have  it  nearer  a  fainter  light  may  be  made 
by  covering  part  of  the  lantern  globe.  This  light 
should  be  faint  enough  to  look  about  like  a  faint 
star,  so  that  no  change  of  focus  of  the  objective  will 
be  required. 

In  good  season  for  beginning  .the  observations, 
set  up  the  plane  table  over  the  instrument  point  of 
the  reference  line,  with  a  sheet  of  paper  on  for  this 
work.  Set  a  needle  at  the  center  of  the  sheet  as  a 
guide  for  the  edge  of  the  rule.  Turn  the  board  so 
there  will  be  room  on  the  paper,  on  the  side  towards 
the  stars  to  be  observed,  and  a  good  place  for  the  plat 
of  the  reference  line.  Clamp  the  spindle.  Set  the 
alidade  on  the  reference  line,  sighting  to  the  distant 
lantern,  the  cross-wires  being  illuminated.  Use  the 


—  55  — 

hands  in  setting  the  alidade,  and  keep  the  edge  of 
the  rule  to  the  needle  at  the  center  of  the  sheet. 
Draw  a  line  from  the  center  of  the  sheet  by  the  edge 
of  the  rule, —  m  r,  Fig.  n.  Place  the  alidade  to  look 
at  a  star,  the  edge  of  the  rule  being  kept  to  the  nee- 
dle. Direct  the  telescope  to  a  star.  Bring  the  star 
into  the  field  of  view  of  the  telescope  by  the  hands. 
Illuminate  the  cross-wires.  Set  the  vertical  wire  on 
the  star,  using  the  hands  and  keeping  the  edge  of  the 
rule  to  the  needle  at  the  center  of  the  sheet.  Keep 
this  wire  on  the  star  very  carefully  as  it  approaches 
its  elongation,  which  may  be  known  by  the  increased 
time  taken  by  the  star  to  cross  the  wire.  Follow  the 
star,  moving  the  alidade  with  the  hands,  keeping 
the  vertical  wire  exactly  on  it  till  it  appears  to  pass  di- 
rectly along  this  wire.  Presently  it  will  leave  this  wire 
and  go  to  the  same  side  it  came  from.  Follow  it  no 
longer  than  till  it  passes  directly  along  the  wire.  Draw 
a  line  from  the  center  of  the  sheet  by  the  edge  of  the 
rule.  Mark  this  line  for  the  star  observed  with  a  note 
of  its  elongation,  east  or  west, —  as  m,  s,  Fig.  n. 
Set  the  alidade  again  on  the  reference  line.  See  if 
m  r  fits  the  rule.  If  it  does  not  fit  well  this  observa- 
tion must  be  rejected.  The  nicety  of  the  fit  depends 
upon  the  character  of  the  work. 

To  save  confusion  on  the  sheet  in  case  the  rule 
does  not  fit  m  r,  set  the  edge  of  the  rule  to  m  r,  and 
set  the  line  of  sight  on  the  distant  lantern,  with  the 
spindle  slow  motion  screw  as  a  preparation  for  the 
next  observation. 

In  case  the  rule  fits  well  enough,  but  not  ex- 
actly, draw  a  second  line  by  the  edge  of  the  rule  and 
mark  both  for  the  star  observed.  If  the  table  stands 


—  56  — 

still -till  time  for  the  next  observation,   mark  the    sec- 
ond line  for  next  star. 


—  57  — 

Observe  a  second  star  as  the  first  was  observed.' 
Draw  a  line  marked  for  it  as  m  s'.  Take  a  pointing 
again  on  the  reference  line  as  before.  "  Mark  the  ref- 
erence line  drawn  on  the  board  if  it  differs  from  the 
last,  —  otherwise  note  the  fact  that  the  lines  checked. 

If  there  are  two  lines  m  r  drawn  on  the  sheet 
for  the  observation  of  a  star,  draw  a  third  line  bisect- 
ing the  angle  between  them  for  the  line  to  be  used  in 
making  the  reductions  which  follow,  unless  there  is 
good  reason  to  take  one  of  the  original  lines  drawn, 
—  if  there  is,  use  that  line. 

Observe  three  or  more  stars  in  this  way,  noting 
the  line  for  each,  and  the  elongation  of  each. 

The  lines  to  the  stars  are  to  be  taken  in  pairs 
and  the  position  of  the  meridian  line  found  graphi- 
cally. There  may  be  as  many  results  found  as  there 
are  pairs  that  can  be  formed  from  the  number  of 
stars  observed.  Three  stars  will  give  3  pairs  and  3 
results.  Four  stars  will  give  6  pairs  and  6  results. 
And  so  on. 

Let  Fig.  1  1  represent  the  lines  needed  in  finding 
the  position,  m  p,  of  the  meridian. 

Take  from  the  Nautical  Almanac  the  apparent 
declinations  of  the  stars  for  the  date  of  the  observa- 
tion. 

If  A0  =  the  azimuth  of  the  star  s  and  80  =its 
apparent  declination,  and  Al  =  the  azimuth  of  the 
star  s'  and  81  =  its  apparent  declination,  the  follow- 
ing equations  may  be  written. 

Tan^(A0+A,)=tan  ^(Ao~Ai)  (cos  S0+cos8,) 

cos  80  —  cos  8l 

—A  )—  tan^  <AQ  +  Ai)  (cos  8Q— 


cos          cos 


—  58  - 

These   equations   may  be  solved   graphically,   being 
simple  proportions. 

Bisect  the  angle  ^  m  s'  by  a  line  m  c.  Lay  off 
m  c  equal  to  ten  parts  of  some  scale,  as  large  as  may 
be.  Draw  ^  s'  perpendicular  to  m  c  at  c.  Draw  two 
indefinite  lines  a  b  and  a  d,  making  any  convenient 
angle  with  each  other.  Lay  off  a  f  an  daf'  = 
natural  cosine  of  80,  or  any  multiple  of  cos  80  (in  this 
case  4  was  used  )  to  the  same  scale  m  c  was  meas- 
ured by.  Lay  off  f  e  equal  to  the  same  multiple  of  cos 
8j,  to  the  same  scale,  and/" e'  =fe.  Lay  off  a  i  = 
s  c  =  s'  c.  Also  i ,  2  =  2,  3  =  3,  4  =  -r  c,  if  desired. 
Join  e  and  e' .  Draw  parallels  to  e  e'  through  i,  2,  3, 
and  4,  intersecting  a  dm  /z15A2,  A3, and  A4;  a  hl  should 
equal  Alt  A2  ;  h^  h^  ;  and  h^  h±.  From  c  on  s  s', 
lay  off  c p  =  a  kl,  towards  the  line  which  marks  the 
star  having  the  greater  declination,  because  the  star 
having  the  greater  declination  will  have  the  smaller 
azimuth  from  the  north  at  its  elongation.  Draw  m  p. 
p  m  r  is  the  angle  between  the  reference  line  and  the 
meridian 

By  taking  the  stars  observed  in  pairs,  a  separate 
result  may  be  obtained  for  each  pair,  and  the  final 
location  of  m  p  made  more  trustworthy.  If  the 
lines  become  too  numerous  and  the  drawing  too  com- 
plicated, separate  drawings  may  be  made  for  each 
pair  by  pricking  the  line  m  r  and  the  lines  for  a 
pair  of  stars  through  onto  a  separate  sheet,  and  do- 
ing the  drawing  on  that  sheet.  After  the  angle  p 
m  r  is  found  it  can  be  compared  with  the  value  on 
the  original  sheet. 

There  are  four  cases  of  this  graphical  solution. 
The  reference  line  m  r  may  be  between  m  s  and  m  s't 


—  59  — 

or  not,  as  in  Fig.  n.  The  stars  may  be  at  the  same 
elongation  or  at  different  elongations,  as  in  Fig.  n. 
In  case  the  stars  are  at  the  same  elongation  the  dis- 
tance corresponding  to  c  p  will  represent  tan  ^ 
(A0  -f-  AJ)  and  must  be  laid  off  from  c  towards  the 
place  of  the  meridian  m  p,  that  is,  to  the  left  of  c  if 
the  'stars  are  both  at  the  eastern  elongation,  or  to 
the  right  of  c  if  at  the  western  elongation,  when  the 
direction  m  p  is  to  the  N.  The  circumstances  of  each 
case  will  dispose  of  all  ambiguity. 

Lay  off  the  denominator  of  the  fraction  in  the 
second  number  of  either  equation  and  the  distance 
corresponding  to  s  c  on  the  same  line.  Lay  off  on 
the  other  line  that  part  of  the  numerator  depending 
for  its  value  upon  80  andSj.  Join  the  points  located 
by  functions  of  80  and  8lf  and  draw  a  parallel  or 
parallels  to  this  line  as  required.  A  multiple  of  the 
cosines  is  taken  in  order  to  bring  e  and  ef  a  good  dis- 
tance apart  s'  c  was  laid  off  four  times  as  a  test  of 
the  precision  of  the  drawing.  The  lines  a  b  and  a  d 
should  be  open  enough  to  give  good  intersections  at 
Aj,  h.2,  etc.,  better  than  in  Fig.  n. 

The  angle  p  m  r  may  be  measured  (34),  and  the 
work  oriented  with  respect  to  the  meridian.  The 
meridian  m  p  may  be  laid  out  on  the  ground  if  de- 
sired. 

The  meridian  may  be  laid  out  on  the  ground  di- 
rectly, without  platting  the  lines  on  the  sheet  if  de- 
sired. After  observing  a  star  at  its  elongation,  point 
the  telescope  downwards  and  s  t  a  stake  marked  for 
the  star  with  a  point  marked  on  it  in  line.  Do  the 
same  for  two  or  more  stars,  setting  the  stakes,  as  far 
away  as  convenient,  yet  near  enough  for  the  points 


—  60  — 

on  them  to  be  well  located.  Reverse  the  telescope 
about  its  axis  ;  shift  the  rule  to  the  other  side  of  the 
needle  at  m  ;  and  level  the  table  anew  after  observing 
£  and  marking  the  line  to  each  star. 
d  ^1  Make  notes  showing  the  position  of 
\2)'  the  telescope  when  observing  each 
star.  In  combining  the  observa- 
tions in  pairs,  place  together  those 
for  two  stars  for  which  the  notes 
show  the  telescope  to  have  been  in 
two  different  positions.  By  so 
doing  the  imperfections  of  the 
instrument  will  do  less  harm. 

If  I  in  Fig.  12  shows  the  place 
of  the  plane  table  C  and  D,  the 
stakes  for  two  stars,  one  at  its  E. 
elongation  and  the  other  at  its  W. 
elongation,  and  I  K  the  meridian, 
make  the  following  measurements 
carefully, —  as  with  a  steel  tape.  C  D  =  d;  I  C  =  a\ 
and  I  D  =  b.  Let  x  =  C  K,  and  y  =  K  D,  the  seg- 
ments of  C  D  between  the  meridian  and  C  and  D. 


=  d 

x  —y  = 


COS30~  b 


a  cos  8    -~  b  cos 


where  80  is  the  apparent  declination  of  the  star  repre- 
sented by  C,  and  8X  the  apparent  declination  of  the 
star  represented  by  D.  Find  x  and  y  and  locate  K 
with  the  plane  table  and  steel  tape.  As  a  check  on 
the  figuring  use 

£ a  cos  80 

y       b  cos  8, 


-01  — 

when    the   stars    are  at    the    same   elongation    d  = 

x  —  y  and 

,        _   ,  a  cos  80  -f-  b  ros  8, 
a  cos  8()  —  £  cos  8  l 

In  this  case  K  will  not  come  between  C  and  D  but  to 
the  right  of  D  if  the  stars  are  in  the  W.  elongation, 
or  to  the  left  of  C  if  in  the  E.  elongation.  Should 
D'  be  set  in  line  from  I  to  D  so  that  I  D'  =  I  C  =  a, 
the  figuring  would  be  simpler.  Measure  C  D'  =  d'  ',  and 

x'—y'  =    -  d'  tan  y2  (80  +  8X)  tan  ^  (8l  —  8J. 
For  the  stars  at  the  same  elongation  d'  =  x'  —y'  and 

xf+y=—d'  cot  %  (80  +  8t)  cot  ^  (80-8i). 
An  inspection  of  this  last  equation  shows  that  stars 
having  nearly   the   same  declination  should  not   be 
used  together,  if  observed  at  the  same  elongation. 

Since  A0  =  C  I  K  and  Al  =  D  I  K,  the  azi- 
muths of  the  stars  at  their  elongations.  C  I  D  =  A0 
-f-  Aj.  For  the  stars  in  the  same  elongation,  C  I  D 
=  A0  —  Aj.  Measure  C  I  D  with  the  plane  table 
or  compute  its  value  from  the  steel  tape  measure- 
ments. Then 


Tan  %  (A0  -  A^  =  -  tan  V2  A0  +  At)  tan  %(«<,  +  *i)  tan  V2  (*0-<i) 
Tan  %  (A0  +Ai)  =  -  tan  V2  (A0  +  AX)  cot  y2  (S0  +  5i)  cot  V2  (*o-«i) 

Observations  on  stars  having  nearly  the  same  decli- 
nation, made  at  the  same  elongation,  should  not  be 
combined. 

The  latitude  of   the   place  <£  may  be   computed 
as  follows  : 

cos  8ft        cos  8, 


cos  <>  = 


LO  —  __ 

sin  A0       sin 


Whence  it  appears  that  the  direction  of  the  me- 
ridian and  the   latitude  of  a    place  may   be    found 


without  reading  angles.     A0    and   A1  may  be  found 
directly  from  the  triangles  C  I  K  and  D  I  K. 

In  case  a  plane  table  is  used  having  compass 
sights  on  its  alidade  the  direction  of  the  meridian 
may  be  found  as  follows  :  Suspend  a  long  plumb  line 
from  a  fixed  support  so  high  that  the  stars  to  be  ob- 
served will  appear  somewhat  below  the  support  Use 
a  large  piece  of  stone  or  some  bricks  for  the  bob,  and 
let  it  hang  in  a  vessel  of  water  to  steady  the  line. 
Drive  two  stakes  some  distance  S.  of  the  plumb  line, 
and  nearly  in  an  E.  and  W.  line.  Saw  off  their  tops 
to  the  same  elevation.  Nail  a  smooth  board  on  top 
of  the  stakes.  The  stakes  should  stand  firm,  at  such 
a  distance  S.  of  the  plumb  line  that  with  the  eye  a 
little  above  the  board,  the  stars  to  be  observed  will 
appear  below  the  support  of  the  plumb  line.  The 
length  of  the  board  should  be  6  or  8  ft.  if  about  20 
ft.  S.  of  plumb  line.  Level  the  board  carefully,  both 
ways.  Remove  one  of  the  compass  sights  from  the 
rule  of  the  alidade.  As  a  star  approaches  its  elonga- 
tion place  the  rule  on  the  board  with  the  compass 
sight  to  the  south,  the  rule  pointing  to  the  plumb 
line.  Looking  through  the  compass  sight  keep  the 
plumb  line  over  the  star  by  shifting  the  alidade  with 
the  hands  on  the  board.  Follow  the  star  till  it  ap- 
pears to  pass  directly  along  the  plumb  line,  and  leave 
it  again  on  the  same  side  it  approached  from.  Illu- 
minate the  plumb  line  if  need  be  by  a  light  from  be- 
hind the  observer.  Follow  the  star  no  longer  than 
till  it  passes  directly  along  the  plumb  line.  Keep  the 
rule  pointing  as  nearly  as  may  be  to  the  plumb  line 
all  the  time.  Mark  the  place  where  the  slit  in  the 
compass  sight  stands  on  the  board  and  number  the 


--63  — 

mark.  Record  in  the  note-book  the  same  number 
with  the  name  of  the  star  and  a  note  of  the  elongation, 
E.  or  W.  Observe  2  or  more  stars  in  this  manner. 
Measure  the  exact  distances  of  each  of  the  marks  on 
the  board  from  the  plumb  line.  Measure  very  fully 
the  distances  of  each  mark  on  the  board  from  all  the 
others.  Record  these  measurements.  Reduce  the 
position  cf  the  meridian  by  calculations  similar  to 
those  above  given,  remembering  that  the  triangle  C 
I  D,  Fig.  12,  lies  S.  of  I.  The  point  K  may  then 
be  marked  on  the  board.  The  line  from  K  on  the 
board  to  the  plumb  line  should  be  very  carefully 
produced  by  repeated  sightings.  This  may  be  done 
with  a  field  glass,  if  one  is  to  be  had,  by  driving  a 
nail  at  K  in  the  board,  standing  back  a  sufficient 
distance,  and  using  the  field  glass  to  see  that  the 
stake  and  nail  beyond  is  set  in  line  of  K  and  the 
plumb  line.  Better  prolong  the  line  some  distance 
S.  of  the  board,  and  then  N.  of  the  plumb  line,  set- 
ting stakes  and  nails  in  both  directions. 

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CHEEVER.— Select  Methods  in  Inorganic  Quantitative  Analysis.  By 
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in  the  University  of-  Michigan.  Revised  and  enlarged  by  Frank 
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aUTHE.— Laboratory  Exercises  with  Primary  and  Storage  Cells 
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CLASSEN-HARRIMAN.—  Quantitative  Analysis.  By  Alexander  Clas- 
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MAR  10  1942 


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